Theorem iscomb 14690

Description: The predicate "is a commutative operation".

Hypothesis

Ref	Expression
iscom.1	|- X = dom dom G

Assertion

Ref	Expression
iscomb	|- ((G e. Com1 /\ A e. X /\ B e. X) -> (AGB) = (BGA))

Proof of Theorem iscomb

Step	Hyp	Ref	Expression
1		iscom.1	. . . . 5 |- X = dom dom G
2	1	iscom 14689	. . . 4 |- (G e. Com1 -> (G e. Com1 <-> A.x e. X A.y e. X (xGy) = (yGx)))
3		opreq1 4889	. . . . . . . 8 |- (x = A -> (xGy) = (AGy))
4		opreq2 4890	. . . . . . . 8 |- (x = A -> (yGx) = (yGA))
5	3, 4	eqeq12d 1899	. . . . . . 7 |- (x = A -> ((xGy) = (yGx) <-> (AGy) = (yGA)))
6		opreq2 4890	. . . . . . . 8 |- (y = B -> (AGy) = (AGB))
7		opreq1 4889	. . . . . . . 8 |- (y = B -> (yGA) = (BGA))
8	6, 7	eqeq12d 1899	. . . . . . 7 |- (y = B -> ((AGy) = (yGA) <-> (AGB) = (BGA)))
9	5, 8	rcla42v 2384	. . . . . 6 |- ((A e. X /\ B e. X) -> (A.x e. X A.y e. X (xGy) = (yGx) -> (AGB) = (BGA)))
10	9	ex 402	. . . . 5 |- (A e. X -> (B e. X -> (A.x e. X A.y e. X (xGy) = (yGx) -> (AGB) = (BGA))))
11	10	com3r 39	. . . 4 |- (A.x e. X A.y e. X (xGy) = (yGx) -> (A e. X -> (B e. X -> (AGB) = (BGA))))
12	2, 11	syl6bi 231	. . 3 |- (G e. Com1 -> (G e. Com1 -> (A e. X -> (B e. X -> (AGB) = (BGA)))))
13	12	pm2.43i 78	. 2 |- (G e. Com1 -> (A e. X -> (B e. X -> (AGB) = (BGA))))
14	13	3imp 1061	1 |- ((G e. Com1 /\ A e. X /\ B e. X) -> (AGB) = (BGA))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  dom cdm 3986  (class class class)co 4884  Com1ccm1 14687

This theorem is referenced by:  reacomsmgrp1 14703  reacomsmgrp2 14704  reacomsmgrp3 14705

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-com1 14688

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