Theorem gacan 19085

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 19072	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
2	1	adantr 481	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp)
3		simpr1 1194	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2736	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		eqid 2736	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		gacan.2	. . . . . . . 8 ⊢ 𝑁 = (invg‘𝐺)
8	4, 5, 6, 7	grprinv 18801	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
9	2, 3, 8	syl2anc 584	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
10	9	oveq1d 7372	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶))
11		simpl 483	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
12	4, 7	grpinvcl 18798	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
13	2, 3, 12	syl2anc 584	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝑁‘𝐴) ∈ 𝑋)
14		simpr3 1196	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
15	4, 5	gaass 19077	. . . . . 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 1372	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)))
17	6	gagrpid 19074	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
18	11, 14, 17	syl2anc 584	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
19	10, 16, 18	3eqtr3d 2784	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) = 𝐶)
20	19	eqeq2d 2747	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ (𝐴 ⊕ 𝐵) = 𝐶))
21		simpr2 1195	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
22	4	gaf 19075	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
23	22	adantr 481	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
24	23, 13, 14	fovcdmd 7526	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝑁‘𝐴) ⊕ 𝐶) ∈ 𝑌)
25	4	galcan 19084	. . . 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 1372	. . 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 2743	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   × cxp 5631  ⟶wf 6492  ‘cfv 6496  (class class class)co 7357  Basecbs 17083  +_gcplusg 17133  0_gc0g 17321  Grpcgrp 18748  inv_gcminusg 18749   GrpAct cga 19069

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-ga 19070

This theorem is referenced by:  gapm  19086  gaorber  19088  gastacl  19089  gastacos  19090