Theorem gacan 19246

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 19233	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
2	1	adantr 480	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp)
3		simpr1 1196	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2737	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		eqid 2737	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		gacan.2	. . . . . . . 8 ⊢ 𝑁 = (invg‘𝐺)
8	4, 5, 6, 7	grprinv 18932	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
9	2, 3, 8	syl2anc 585	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
10	9	oveq1d 7383	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶))
11		simpl 482	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
12	4, 7	grpinvcl 18929	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
13	2, 3, 12	syl2anc 585	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝑁‘𝐴) ∈ 𝑋)
14		simpr3 1198	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
15	4, 5	gaass 19238	. . . . . 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 1375	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)))
17	6	gagrpid 19235	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
18	11, 14, 17	syl2anc 585	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
19	10, 16, 18	3eqtr3d 2780	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) = 𝐶)
20	19	eqeq2d 2748	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ (𝐴 ⊕ 𝐵) = 𝐶))
21		simpr2 1197	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
22	4	gaf 19236	. . . . . 6 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
23	22	adantr 480	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ :(𝑋 × 𝑌)⟶𝑌)
24	23, 13, 14	fovcdmd 7540	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝑁‘𝐴) ⊕ 𝐶) ∈ 𝑌)
25	4	galcan 19245	. . . 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 1375	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
27	20, 26	bitr3d 281	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
28		eqcom 2744	. 2 ⊢ (𝐵 = ((𝑁‘𝐴) ⊕ 𝐶) ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵)
29	27, 28	bitrdi 287	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   × cxp 5630  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  0_gc0g 17371  Grpcgrp 18875  inv_gcminusg 18876   GrpAct cga 19230

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-ga 19231

This theorem is referenced by:  gapm  19247  gaorber  19249  gastacl  19250  gastacos  19251