Theorem gacan 19294

Description: Group inverses cancel in a group action. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypotheses

Ref	Expression
galcan.1	⊢ 𝑋 = (Base‘𝐺)
gacan.2	⊢ 𝑁 = (invg‘𝐺)

Assertion

Ref	Expression
gacan	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵))

Proof of Theorem gacan

Step	Hyp	Ref	Expression
1		gagrp 19281	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
2	1	adantr 479	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp)
3		simpr1 1191	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2725	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		eqid 2725	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		gacan.2	. . . . . . . 8 ⊢ 𝑁 = (invg‘𝐺)
8	4, 5, 6, 7	grprinv 18980	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
9	2, 3, 8	syl2anc 582	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
10	9	oveq1d 7438	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶))
11		simpl 481	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
12	4, 7	grpinvcl 18977	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
13	2, 3, 12	syl2anc 582	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝑁‘𝐴) ∈ 𝑋)
14		simpr3 1193	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
15	4, 5	gaass 19286	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ (𝑁‘𝐴) ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)))
16	11, 3, 13, 14, 15	syl13anc 1369	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)))
17	6	gagrpid 19283	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
18	11, 14, 17	syl2anc 582	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
19	10, 16, 18	3eqtr3d 2773	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) = 𝐶)
20	19	eqeq2d 2736	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ (𝐴 ⊕ 𝐵) = 𝐶))
21		simpr2 1192	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
22	4	gaf 19284	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
23	22	adantr 479	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
24	23, 13, 14	fovcdmd 7597	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝑁‘𝐴) ⊕ 𝐶) ∈ 𝑌)
25	4	galcan 19293	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ ((𝑁‘𝐴) ⊕ 𝐶) ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
26	11, 3, 21, 24, 25	syl13anc 1369	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
27	20, 26	bitr3d 280	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
28		eqcom 2732	. 2 ⊢ (𝐵 = ((𝑁‘𝐴) ⊕ 𝐶) ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵)
29	27, 28	bitrdi 286	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   × cxp 5679  ⟶wf 6549  ‘cfv 6553  (class class class)co 7423  Basecbs 17208  +_gcplusg 17261  0_gc0g 17449  Grpcgrp 18923  inv_gcminusg 18924   GrpAct cga 19278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pow 5368  ax-pr 5432  ax-un 7745

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5579  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7379  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8856  df-0g 17451  df-mgm 18628  df-sgrp 18707  df-mnd 18723  df-grp 18926  df-minusg 18927  df-ga 19279

This theorem is referenced by:  gapm  19295  gaorber  19297  gastacl  19298  gastacos  19299