Theorem gacan 19255

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 19242	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
2	1	adantr 480	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp)
3		simpr1 1192	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2728	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		eqid 2728	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		gacan.2	. . . . . . . 8 ⊢ 𝑁 = (invg‘𝐺)
8	4, 5, 6, 7	grprinv 18946	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
9	2, 3, 8	syl2anc 583	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
10	9	oveq1d 7435	. . . . 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 18943	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
13	2, 3, 12	syl2anc 583	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝑁‘𝐴) ∈ 𝑋)
14		simpr3 1194	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
15	4, 5	gaass 19247	. . . . . 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 1370	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)))
17	6	gagrpid 19244	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
18	11, 14, 17	syl2anc 583	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
19	10, 16, 18	3eqtr3d 2776	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) = 𝐶)
20	19	eqeq2d 2739	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ (𝐴 ⊕ 𝐵) = 𝐶))
21		simpr2 1193	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
22	4	gaf 19245	. . . . . 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 7593	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝑁‘𝐴) ⊕ 𝐶) ∈ 𝑌)
25	4	galcan 19254	. . . 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 1370	. . 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 2735	. 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 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   × cxp 5676  ⟶wf 6544  ‘cfv 6548  (class class class)co 7420  Basecbs 17179  +_gcplusg 17232  0_gc0g 17420  Grpcgrp 18889  inv_gcminusg 18890   GrpAct cga 19239

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-map 8846  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18892  df-minusg 18893  df-ga 19240

This theorem is referenced by:  gapm  19256  gaorber  19258  gastacl  19259  gastacos  19260