:: Field Properties of Complex Numbers - Requirements
:: by Library Committee
::
:: Received May 29, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies XCMPLX_0, ARYTM_3, CARD_1, NUMBERS, SUBSET_1, ARYTM_0, ARYTM_1,
RELAT_1;
notations TARSKI, SUBSET_1, ORDINAL1, NUMBERS, ARYTM_0, XCMPLX_0;
constructors FUNCT_4, ARYTM_0, XCMPLX_0, NUMBERS;
registrations XCMPLX_0, ORDINAL1;
requirements NUMERALS, SUBSET, BOOLE;
theorems ARYTM_0, XCMPLX_0, NUMBERS;
begin
:: This file contains statements which are obvious for Mizar checker if
:: "requirements ARITHM" is included in the environment description
:: of an article. "requirements NUMERALS" is also required.
:: They are published for testing purposes only.
:: Users should use appropriate requirements instead of referencing
:: to these theorems.
:: Some of these items need also other requirements for proper work.
reserve x for Complex;
theorem Th1:
x + 0 = x
proof
0 in NAT;
then reconsider Z =0 as Element of REAL by NUMBERS:19;
x in COMPLEX by XCMPLX_0:def 2;
then consider x1,x2 being Element of REAL such that
A1: x = [*x1,x2*] by ARYTM_0:9;
0 = [*Z,Z*] by ARYTM_0:def 5;
then x + 0 = [*+(x1,Z),+(x2,Z)*] by A1,XCMPLX_0:def 4
.= [* x1,+(x2,Z)*] by ARYTM_0:11
.= x by A1,ARYTM_0:11;
hence thesis;
end;
Lm1: -0 = 0
proof
0 + -0 = -0 by Th1;
hence thesis by XCMPLX_0:def 6;
end;
theorem Th2:
x * 0 = 0
proof
0 in NAT;
then reconsider Z =0 as Element of REAL by NUMBERS:19;
x in COMPLEX by XCMPLX_0:def 2;
then consider x1,x2 being Element of REAL such that
A1: x = [*x1,x2*] by ARYTM_0:9;
+(Z,Z) = 0 by ARYTM_0:11;
then Lm2: opp Z = 0 by ARYTM_0:def 3;
0 = [*Z,Z*] by ARYTM_0:def 5;
then x * 0 = [* +(*(x1,Z),opp*(x2,Z)), +(*(x1,Z),*(x2,Z)) *] by A1,
XCMPLX_0:def 5
.= [* +(*(x1,Z),opp Z), +(*(x1,Z),*(x2,Z)) *] by ARYTM_0:12
.= [* +(*(x1,Z),opp Z), +(*(x1,Z),Z) *] by ARYTM_0:12
.= [* +(Z,opp Z), +(*(x1,Z),Z) *] by ARYTM_0:12
.= [* +(Z,opp Z), +(Z,Z) *] by ARYTM_0:12
.= [* +(Z,opp Z), Z *] by ARYTM_0:11
.= [* opp Z, Z *] by ARYTM_0:11
.= 0 by Lm2,ARYTM_0:def 5;
hence thesis;
end;
theorem Th3:
1 * x = x
proof
0 in NAT & 1 in NAT;
then reconsider Z =0,J = 1 as Element of REAL by NUMBERS:19;
+(Z,Z) = 0 by ARYTM_0:11;
then Lm2: opp Z = 0 by ARYTM_0:def 3;
x in COMPLEX by XCMPLX_0:def 2;
then consider x1,x2 being Element of REAL such that
A1: x = [*x1,x2*] by ARYTM_0:9;
1 = [*J,Z*] by ARYTM_0:def 5;
then x * 1 = [* +(*(x1,J),opp*(x2,Z)), +(*(x1,Z),*(x2,J)) *] by A1,
XCMPLX_0:def 5
.= [* +(*(x1,J),opp Z), +(*(x1,Z),*(x2,J)) *] by ARYTM_0:12
.= [* +(x1,opp Z), +(*(x1,Z),*(x2,J)) *] by ARYTM_0:19
.= [* +(x1,opp Z), +(*(x1,Z),x2) *] by ARYTM_0:19
.= [* +(x1,Z), +(Z,x2) *] by Lm2,ARYTM_0:12
.= [* x1, +(Z,x2) *] by ARYTM_0:11
.= x by A1,ARYTM_0:11;
hence thesis;
end;
theorem
x - 0 = x
proof
x - 0 = x + 0 by Lm1,XCMPLX_0:def 8;
hence thesis by Th1;
end;
theorem
0 / x = 0
proof
0 / x = 0 * x" by XCMPLX_0:def 9;
hence thesis by Th2;
end;
Lm3: 1" = 1
proof
1 * 1" = 1" by Th3;
hence thesis by XCMPLX_0:def 7;
end;
theorem
x / 1 = x
proof
x / 1 = x * 1 by Lm3,XCMPLX_0:def 9;
hence thesis by Th3;
end;