Theorem galcan 19168

Description: The action of a particular group element is left-cancelable. (Contributed by FL, 17-May-2010.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypothesis

Ref	Expression
galcan.1	⊢ 𝑋 = (Base‘𝐺)

Assertion

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

Proof of Theorem galcan

Step	Hyp	Ref	Expression
1		oveq2 7417	. . 3 ⊢ ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)))
2		simpl 484	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
3		gagrp 19156	. . . . . . . 8 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → 𝐺 ∈ Grp)
4	2, 3	syl 17	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐺 ∈ Grp)
5		simpr1 1195	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
6		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
7		eqid 2733	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2733	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
9		eqid 2733	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	6, 7, 8, 9	grplinv 18874	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺))
11	4, 5, 10	syl2anc 585	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺))
12	11	oveq1d 7424	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = ((0g‘𝐺) ⊕ 𝐵))
13	6, 9	grpinvcl 18872	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
14	4, 5, 13	syl2anc 585	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
15		simpr2 1196	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
16	6, 7	gaass 19161	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)))
17	2, 14, 5, 15, 16	syl13anc 1373	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)))
18	8	gagrpid 19158	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵)
19	2, 15, 18	syl2anc 585	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵)
20	12, 17, 19	3eqtr3d 2781	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = 𝐵)
21	11	oveq1d 7424	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶))
22		simpr3 1197	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
23	6, 7	gaass 19161	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (((invg‘𝐺)‘𝐴) ∈ 𝑋 ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)))
24	2, 14, 5, 22, 23	syl13anc 1373	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)))
25	8	gagrpid 19158	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
26	2, 22, 25	syl2anc 585	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
27	21, 24, 26	3eqtr3d 2781	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) = 𝐶)
28	20, 27	eqeq12d 2749	. . 3 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) ↔ 𝐵 = 𝐶))
29	1, 28	imbitrid 243	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → 𝐵 = 𝐶))
30		oveq2 7417	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶))
31	29, 30	impbid1 224	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  +_gcplusg 17197  0_gc0g 17385  Grpcgrp 18819  inv_gcminusg 18820   GrpAct cga 19153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-ga 19154

This theorem is referenced by:  gacan  19169