Theorem gacan 17738

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 17725	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
2	1	adantr 481	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp)
3		simpr1 1067	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
4		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
5		eqid 2622	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
6		eqid 2622	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
7		gacan.2	. . . . . . . 8 ⊢ 𝑁 = (invg‘𝐺)
8	4, 5, 6, 7	grprinv 17469	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
9	2, 3, 8	syl2anc 693	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴(+g‘𝐺)(𝑁‘𝐴)) = (0g‘𝐺))
10	9	oveq1d 6665	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶))
11		simpl 473	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
12	4, 7	grpinvcl 17467	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ 𝑋)
13	2, 3, 12	syl2anc 693	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝑁‘𝐴) ∈ 𝑋)
14		simpr3 1069	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
15	4, 5	gaass 17730	. . . . . 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 1328	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴(+g‘𝐺)(𝑁‘𝐴)) ⊕ 𝐶) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)))
17	6	gagrpid 17727	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
18	11, 14, 17	syl2anc 693	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
19	10, 16, 18	3eqtr3d 2664	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) = 𝐶)
20	19	eqeq2d 2632	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ (𝐴 ⊕ 𝐵) = 𝐶))
21		simpr2 1068	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
22	4	gaf 17728	. . . . . 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	fovrnd 6806	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝑁‘𝐴) ⊕ 𝐶) ∈ 𝑌)
25	4	galcan 17737	. . . 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 1328	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ ((𝑁‘𝐴) ⊕ 𝐶)) ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
27	20, 26	bitr3d 270	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ 𝐵 = ((𝑁‘𝐴) ⊕ 𝐶)))
28		eqcom 2629	. 2 ⊢ (𝐵 = ((𝑁‘𝐴) ⊕ 𝐶) ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵)
29	27, 28	syl6bb 276	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = 𝐶 ↔ ((𝑁‘𝐴) ⊕ 𝐶) = 𝐵))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Grpcgrp 17422  inv_gcminusg 17423   GrpAct cga 17722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425  df-minusg 17426  df-ga 17723

This theorem is referenced by:  gapm  17739  gaorber  17741  gastacl  17742  gastacos  17743