:: Uniqueness of factoring an integer and multiplicative group $Z/pZ^{*}$
:: by Hiroyuki Okazaki and Yasunari Shidama
::
:: Received January 31, 2008
:: Copyright (c) 2008 Association of Mizar Users
Lm1:
for x, X, y, z being set st z <> x holds
((X --> 0 ) +* x,y) . z = 0
theorem Th1: :: INT_7:1
theorem Th2: :: INT_7:2
theorem Th3: :: INT_7:3
theorem Th4: :: INT_7:4
Lm2:
for p, q being bag of SetPrimes st support p c= support q & p | (support p) = q | (support p) holds
ex r being bag of SetPrimes st
( support r = (support q) \ (support p) & support p misses support r & r | (support r) = q | (support r) & p + r = q )
Lm3:
for p being bag of SetPrimes
for X being set st X c= support p holds
ex q, r being bag of SetPrimes st
( support q = (support p) \ X & support r = X & support q misses support r & q | (support q) = p | (support q) & r | (support r) = p | (support r) & q + r = p )
:: deftheorem Def1 defines prime-factorization-like INT_7:def 1 :
theorem Th5: :: INT_7:5
Lm4:
for a being Prime
for b being bag of SetPrimes st b is prime-factorization-like & a in support b holds
( a divides Product b & ex n being natural number st a |^ n divides Product b )
Lm5:
for p being FinSequence of NAT
for x being Element of NAT
for b being bag of SetPrimes st b is prime-factorization-like & p ^ <*x*> = b * (canFS (support b)) holds
ex p1 being FinSequence of NAT ex q being Prime ex n being Element of NAT ex b1 being bag of SetPrimes st
( 0 < n & b1 is prime-factorization-like & q in support b & support b1 = (support b) \ {q} & x = q |^ n & len p1 = len p & Product p = Product p1 & p1 = b1 * (canFS (support b1)) )
Lm6:
for i being Element of NAT
for f being FinSequence of NAT
for b being bag of SetPrimes
for a being Prime st len f = i & b is prime-factorization-like & Product b <> 1 & a divides Product b & Product b = Product f & f = b * (canFS (support b)) holds
a in support b
theorem Th6: :: INT_7:6
Lm7:
for a being Prime
for b being bag of SetPrimes st b is prime-factorization-like & a divides Product b holds
a in support b
theorem Th7: :: INT_7:7
theorem :: INT_7:8
theorem Th9: :: INT_7:9
theorem Th10: :: INT_7:10
theorem :: INT_7:11
theorem Th12: :: INT_7:12
Lm8:
for q being Prime
for g being Element of NAT st g <> 0 holds
ex p1 being bag of SetPrimes st
( p1 = (SetPrimes --> 0 ) +* q,g & support p1 = {q} & Product p1 = g )
Lm9:
for p being bag of SetPrimes
for x being Prime st p is prime-factorization-like & x in support p & p . x = x holds
ex p1, r1 being bag of SetPrimes st
( p1 is prime-factorization-like & support r1 = {x} & Product r1 = x & support p1 = (support p) \ {x} & p1 | (support p1) = p | (support p1) & Product p = (Product p1) * x )
Lm10:
for p being bag of SetPrimes
for x being Prime st p is prime-factorization-like & x in support p & p . x <> x holds
ex p1, r1 being bag of SetPrimes st
( p1 is prime-factorization-like & support r1 = {x} & Product r1 = x & support p1 = support p & p1 | ((support p1) \ {x}) = p | ((support p1) \ {x}) & p . x = (p1 . x) * x & Product p = (Product p1) * x )
theorem Th13: :: INT_7:13
theorem Th14: :: INT_7:14
Lm11:
for n being Element of NAT
for p, q being bag of SetPrimes st p is prime-factorization-like & q is prime-factorization-like & Product p <= 2 |^ n & Product p = Product q holds
p = q
theorem Th15: :: INT_7:15
theorem :: INT_7:16
theorem Th17: :: INT_7:17
:: deftheorem Def2 defines Segm0 INT_7:def 2 :
theorem Th18: :: INT_7:18
:: deftheorem defines multint0 INT_7:def 3 :
Lm12:
for p being Prime
for a, b being Element of multMagma(# (Segm0 p),(multint0 p) #)
for x, y being Element of (INT.Ring p) st a = x & y = b holds
x * y = a * b
theorem Th19: :: INT_7:19
:: deftheorem defines Z/Z* INT_7:def 4 :
theorem :: INT_7:20
theorem Th21: :: INT_7:21
theorem :: INT_7:22
theorem :: INT_7:23
theorem :: INT_7:24
theorem Th25: :: INT_7:25
Lm13:
for L being Field
for n being Element of NAT
for f being non-zero Polynomial of L st deg f = n holds
ex m being Element of NAT st
( m = card (Roots f) & m <= n )
theorem Th26: :: INT_7:26
theorem Th27: :: INT_7:27
theorem Th28: :: INT_7:28
Lm14:
for p being Prime
for L being Field
for n being natural number st 0 < n & L = INT.Ring p holds
ex f being Polynomial of L st
( deg f = n & ( for x being Element of L
for xz being Element of (Z/Z* p)
for xn being Element of (INT.Ring p) st x = xz & xn = xz |^ n holds
eval f,x = xn - (1_ (INT.Ring p)) ) )
theorem Th29: :: INT_7:29
theorem Th30: :: INT_7:30
theorem :: INT_7:31