Theorem gaplc 14731

Description: The action of a particular group element is left-cancelable.

Hypotheses

Ref	Expression
gaplc.1	|- X = ran G
gaplc.2	|- Y = ran M

Assertion

Ref	Expression
gaplc	|- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((CMA) = (CMB) -> A = B))

Proof of Theorem gaplc

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- ran G = ran G
2		eqid 1884	. . . 4 |- ran M = ran M
3		eqid 1884	. . . 4 |- (M o. `'(2nd |` ({C} X. _V))) = (M o. `'(2nd |` ({C} X. _V)))
4	1, 2, 3	gapm 9462	. . 3 |- ((M e. N /\ <.G, M>. e. GrpAct /\ C e. ran G) -> (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M)
5		gaplc.1	. . . . . 6 |- X = ran G
6	5	eleq2i 1961	. . . . 5 |- (C e. X <-> C e. ran G)
7	6	biimpi 168	. . . 4 |- (C e. X -> C e. ran G)
8	7	3ad2ant1 897	. . 3 |- ((C e. X /\ A e. Y /\ B e. Y) -> C e. ran G)
9	4, 8	syl3an3 1132	. 2 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M)
10		gagrp 9456	. . . . . . . . 9 |- ((M e. N /\ <.G, M>. e. GrpAct) -> G e. Grp)
11	5	unsgrp 14728	. . . . . . . . . 10 |- (G e. Grp -> X e. _V)
12		rnexg 4207	. . . . . . . . . . . 12 |- (M e. N -> ran M e. _V)
13		gaplc.2	. . . . . . . . . . . 12 |- Y = ran M
14	12, 13	syl5eqel 1975	. . . . . . . . . . 11 |- (M e. N -> Y e. _V)
15	14	adantr 425	. . . . . . . . . 10 |- ((M e. N /\ <.G, M>. e. GrpAct) -> Y e. _V)
16	11, 15	anim12i 360	. . . . . . . . 9 |- ((G e. Grp /\ (M e. N /\ <.G, M>. e. GrpAct)) -> (X e. _V /\ Y e. _V))
17	10, 16	mpancom 769	. . . . . . . 8 |- ((M e. N /\ <.G, M>. e. GrpAct) -> (X e. _V /\ Y e. _V))
18	17	3adant3 896	. . . . . . 7 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (X e. _V /\ Y e. _V))
19		ne0i 2881	. . . . . . . . . 10 |- (A e. Y -> Y =/= (/))
20	19	3ad2ant2 898	. . . . . . . . 9 |- ((C e. X /\ A e. Y /\ B e. Y) -> Y =/= (/))
21	20	3ad2ant3 899	. . . . . . . 8 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> Y =/= (/))
22	1, 2	gaf 9457	. . . . . . . . . 10 |- ((M e. N /\ <.G, M>. e. GrpAct) -> M:(ran G X. ran M)-->ran M)
23		ffn 4562	. . . . . . . . . . 11 |- (M:(ran G X. ran M)-->ran M -> M Fn (ran G X. ran M))
24		xpeq1 4016	. . . . . . . . . . . . . . 15 |- (X = ran G -> (X X. ran M) = (ran G X. ran M))
25	24	eqcomd 1889	. . . . . . . . . . . . . 14 |- (X = ran G -> (ran G X. ran M) = (X X. ran M))
26	5, 25	ax-mp 7	. . . . . . . . . . . . 13 |- (ran G X. ran M) = (X X. ran M)
27		xpeq2 4017	. . . . . . . . . . . . . . 15 |- (Y = ran M -> (X X. Y) = (X X. ran M))
28	27	eqcomd 1889	. . . . . . . . . . . . . 14 |- (Y = ran M -> (X X. ran M) = (X X. Y))
29	13, 28	ax-mp 7	. . . . . . . . . . . . 13 |- (X X. ran M) = (X X. Y)
30	26, 29	eqtri 1908	. . . . . . . . . . . 12 |- (ran G X. ran M) = (X X. Y)
31		fneq2 4504	. . . . . . . . . . . . 13 |- ((ran G X. ran M) = (X X. Y) -> (M Fn (ran G X. ran M) <-> M Fn (X X. Y)))
32	31	biimpd 170	. . . . . . . . . . . 12 |- ((ran G X. ran M) = (X X. Y) -> (M Fn (ran G X. ran M) -> M Fn (X X. Y)))
33	30, 32	ax-mp 7	. . . . . . . . . . 11 |- (M Fn (ran G X. ran M) -> M Fn (X X. Y))
34	23, 33	syl 12	. . . . . . . . . 10 |- (M:(ran G X. ran M)-->ran M -> M Fn (X X. Y))
35	22, 34	syl 12	. . . . . . . . 9 |- ((M e. N /\ <.G, M>. e. GrpAct) -> M Fn (X X. Y))
36	35	3adant3 896	. . . . . . . 8 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> M Fn (X X. Y))
37	21, 36	jca 310	. . . . . . 7 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (Y =/= (/) /\ M Fn (X X. Y)))
38		3simpa 872	. . . . . . . 8 |- ((C e. X /\ A e. Y /\ B e. Y) -> (C e. X /\ A e. Y))
39	38	3ad2ant3 899	. . . . . . 7 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (C e. X /\ A e. Y))
40	18, 37, 39	3jca 1050	. . . . . 6 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ A e. Y)))
41		3simpb 873	. . . . . . . 8 |- ((C e. X /\ A e. Y /\ B e. Y) -> (C e. X /\ B e. Y))
42	41	3ad2ant3 899	. . . . . . 7 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (C e. X /\ B e. Y))
43	18, 37, 42	3jca 1050	. . . . . 6 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ B e. Y)))
44	40, 43	jca 310	. . . . 5 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ A e. Y)) /\ ((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ B e. Y))))
45	44	adantr 425	. . . 4 |- (((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) /\ (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M) -> (((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ A e. Y)) /\ ((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ B e. Y))))
46		valvalcurfn 14554	. . . . 5 |- (((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ A e. Y)) -> (((cur1` M)` C)` A) = (CMA))
47		valvalcurfn 14554	. . . . 5 |- (((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ B e. Y)) -> (((cur1` M)` C)` B) = (CMB))
48	46, 47	anim12i 360	. . . 4 |- ((((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ A e. Y)) /\ ((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ (C e. X /\ B e. Y))) -> ((((cur1` M)` C)` A) = (CMA) /\ (((cur1` M)` C)` B) = (CMB)))
49	45, 48	syl 12	. . 3 |- (((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) /\ (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M) -> ((((cur1` M)` C)` A) = (CMA) /\ (((cur1` M)` C)` B) = (CMB)))
50		eqcom 1886	. . . . . . 7 |- ((((cur1` M)` C)` A) = (CMA) <-> (CMA) = (((cur1` M)` C)` A))
51	50	biimpi 168	. . . . . 6 |- ((((cur1` M)` C)` A) = (CMA) -> (CMA) = (((cur1` M)` C)` A))
52		eqcom 1886	. . . . . . 7 |- ((((cur1` M)` C)` B) = (CMB) <-> (CMB) = (((cur1` M)` C)` B))
53	52	biimpi 168	. . . . . 6 |- ((((cur1` M)` C)` B) = (CMB) -> (CMB) = (((cur1` M)` C)` B))
54	51, 53	eqeqan12d 1901	. . . . 5 |- (((((cur1` M)` C)` A) = (CMA) /\ (((cur1` M)` C)` B) = (CMB)) -> ((CMA) = (CMB) <-> (((cur1` M)` C)` A) = (((cur1` M)` C)` B)))
55	54	biimpd 170	. . . 4 |- (((((cur1` M)` C)` A) = (CMA) /\ (((cur1` M)` C)` B) = (CMB)) -> ((CMA) = (CMB) -> (((cur1` M)` C)` A) = (((cur1` M)` C)` B)))
56		simp31 912	. . . . . . 7 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> C e. X)
57	3	valcurfn1 14552	. . . . . . 7 |- (((X e. _V /\ Y e. _V) /\ (Y =/= (/) /\ M Fn (X X. Y)) /\ C e. X) -> ((cur1` M)` C) = (M o. `'(2nd |` ({C} X. _V))))
58	18, 21, 36, 56, 57	syl121anc 1105	. . . . . 6 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((cur1` M)` C) = (M o. `'(2nd |` ({C} X. _V))))
59		f1oeq1 4630	. . . . . . . 8 |- (((cur1` M)` C) = (M o. `'(2nd |` ({C} X. _V))) -> (((cur1` M)` C):ran M-1-1-onto->ran M <-> (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M))
60		f1oeq2 4631	. . . . . . . . . . . 12 |- (ran M = Y -> (((cur1` M)` C):ran M-1-1-onto->ran M <-> ((cur1` M)` C):Y-1-1-onto->ran M))
61		f1oeq3 4632	. . . . . . . . . . . 12 |- (ran M = Y -> (((cur1` M)` C):Y-1-1-onto->ran M <-> ((cur1` M)` C):Y-1-1-onto->Y))
62	60, 61	bitrd 587	. . . . . . . . . . 11 |- (ran M = Y -> (((cur1` M)` C):ran M-1-1-onto->ran M <-> ((cur1` M)` C):Y-1-1-onto->Y))
63		f1fveq 4852	. . . . . . . . . . . . . . . 16 |- ((((cur1` M)` C):Y-1-1->Y /\ (A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) <-> A = B))
64	63	biimpd 170	. . . . . . . . . . . . . . 15 |- ((((cur1` M)` C):Y-1-1->Y /\ (A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B))
65	64	expcom 403	. . . . . . . . . . . . . 14 |- ((A e. Y /\ B e. Y) -> (((cur1` M)` C):Y-1-1->Y -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B)))
66	65	3adant1 894	. . . . . . . . . . . . 13 |- ((C e. X /\ A e. Y /\ B e. Y) -> (((cur1` M)` C):Y-1-1->Y -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B)))
67	66	3ad2ant3 899	. . . . . . . . . . . 12 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> (((cur1` M)` C):Y-1-1->Y -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B)))
68		f1of1 4634	. . . . . . . . . . . 12 |- (((cur1` M)` C):Y-1-1-onto->Y -> ((cur1` M)` C):Y-1-1->Y)
69	67, 68	syl5com 63	. . . . . . . . . . 11 |- (((cur1` M)` C):Y-1-1-onto->Y -> ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B)))
70	62, 69	syl6bi 231	. . . . . . . . . 10 |- (ran M = Y -> (((cur1` M)` C):ran M-1-1-onto->ran M -> ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B))))
71	70	eqcoms 1887	. . . . . . . . 9 |- (Y = ran M -> (((cur1` M)` C):ran M-1-1-onto->ran M -> ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B))))
72	13, 71	ax-mp 7	. . . . . . . 8 |- (((cur1` M)` C):ran M-1-1-onto->ran M -> ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B)))
73	59, 72	syl6bir 232	. . . . . . 7 |- (((cur1` M)` C) = (M o. `'(2nd |` ({C} X. _V))) -> ((M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M -> ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B))))
74	73	com23 36	. . . . . 6 |- (((cur1` M)` C) = (M o. `'(2nd |` ({C} X. _V))) -> ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B))))
75	58, 74	mpcom 60	. . . . 5 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B)))
76	75	imp 377	. . . 4 |- (((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) /\ (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M) -> ((((cur1` M)` C)` A) = (((cur1` M)` C)` B) -> A = B))
77	55, 76	syl9r 72	. . 3 |- (((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) /\ (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M) -> (((((cur1` M)` C)` A) = (CMA) /\ (((cur1` M)` C)` B) = (CMB)) -> ((CMA) = (CMB) -> A = B)))
78	49, 77	mpd 29	. 2 |- (((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) /\ (M o. `'(2nd |` ({C} X. _V))):ran M-1-1-onto->ran M) -> ((CMA) = (CMB) -> A = B))
79	9, 78	mpdan 768	1 |- ((M e. N /\ <.G, M>. e. GrpAct /\ (C e. X /\ A e. Y /\ B e. Y)) -> ((CMA) = (CMB) -> A = B))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  _Vcvv 2292  (/)c0 2875  {csn 3044  <.cop 3046   X. cxp 3984  `'ccnv 3985  ran crn 3987   |` cres 3988   o. ccom 3990   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -1-1-onto->wf1o 3997  ` cfv 3998  (class class class)co 4884  2ndc2nd 5019  Grpcgr 9311  GrpActcga 9447  cur1ccur1 14542

This theorem is referenced by:  curgrpact 14735

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-rep 3428  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-if 2983  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-1st 5020  df-2nd 5021  df-grp 9316  df-gid 9317  df-ginv 9318  df-ga 9448  df-cur1 14544

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