Theorem reacomsmgrp4 14706

Description: Rearrangement of terms in a commutative semi-group.

Hypothesis

Ref	Expression
reacomsmgrp1.1	|- X = dom dom G

Assertion

Ref	Expression
reacomsmgrp4	|- ((G e. (SemiGrp i^i Com1) /\ (A e. X /\ B e. X /\ C e. X)) -> (AG(BGC)) = ((CGB)GA))

Proof of Theorem reacomsmgrp4

Step	Hyp	Ref	Expression
1		reacomsmgrp1.1	. . 3 |- X = dom dom G
2	1	reacomsmgrp2 14704	. 2 |- ((G e. (SemiGrp i^i Com1) /\ (A e. X /\ B e. X /\ C e. X)) -> (AG(BGC)) = (CG(BGA)))
3	1	smgrpass2 14701	. . . 4 |- ((G e. SemiGrp /\ (C e. X /\ B e. X /\ A e. X)) -> ((CGB)GA) = (CG(BGA)))
4	3	eqcomd 1889	. . 3 |- ((G e. SemiGrp /\ (C e. X /\ B e. X /\ A e. X)) -> (CG(BGA)) = ((CGB)GA))
5		inss1 2812	. . . 4 |- (SemiGrp i^i Com1) C_ SemiGrp
6	5	sseli 2617	. . 3 |- (G e. (SemiGrp i^i Com1) -> G e. SemiGrp)
7		simp3 878	. . . 4 |- ((A e. X /\ B e. X /\ C e. X) -> C e. X)
8		simp2 877	. . . 4 |- ((A e. X /\ B e. X /\ C e. X) -> B e. X)
9		simp1 876	. . . 4 |- ((A e. X /\ B e. X /\ C e. X) -> A e. X)
10	7, 8, 9	3jca 1050	. . 3 |- ((A e. X /\ B e. X /\ C e. X) -> (C e. X /\ B e. X /\ A e. X))
11	4, 6, 10	syl2an 503	. 2 |- ((G e. (SemiGrp i^i Com1) /\ (A e. X /\ B e. X /\ C e. X)) -> (CG(BGA)) = ((CGB)GA))
12	2, 11	eqtrd 1925	1 |- ((G e. (SemiGrp i^i Com1) /\ (A e. X /\ B e. X /\ C e. X)) -> (AG(BGC)) = ((CGB)GA))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   i^i cin 2592  dom cdm 3986  (class class class)co 4884  SemiGrpcsem 10377  Com1ccm1 14687

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-13 1311  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  ax-un 3790

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-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-ass 10360  df-mgm 10366  df-sgr 10378  df-com1 14688

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