Theorem galcan 19231

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 7364	. . 3 ⊢ ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)))
2		simpl 482	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
3		gagrp 19219	. . . . . . . 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 2734	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2734	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
9		eqid 2734	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	6, 7, 8, 9	grplinv 18917	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺))
11	4, 5, 10	syl2anc 584	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺))
12	11	oveq1d 7371	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = ((0g‘𝐺) ⊕ 𝐵))
13	6, 9	grpinvcl 18915	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
14	4, 5, 13	syl2anc 584	. . . . . 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 19224	. . . . . 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 1374	. . . . 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 19221	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵)
19	2, 15, 18	syl2anc 584	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵)
20	12, 17, 19	3eqtr3d 2777	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = 𝐵)
21	11	oveq1d 7371	. . . . 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 19224	. . . . . 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 1374	. . . . 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 19221	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
26	2, 22, 25	syl2anc 584	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
27	21, 24, 26	3eqtr3d 2777	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) = 𝐶)
28	20, 27	eqeq12d 2750	. . 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 244	. 2 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → 𝐵 = 𝐶))
30		oveq2 7364	. 2 ⊢ (𝐵 = 𝐶 → (𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶))
31	29, 30	impbid1 225	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) ↔ 𝐵 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ‘cfv 6490  (class class class)co 7356  Basecbs 17134  +_gcplusg 17175  0_gc0g 17357  Grpcgrp 18861  inv_gcminusg 18862   GrpAct cga 19216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-map 8763  df-0g 17359  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-grp 18864  df-minusg 18865  df-ga 19217

This theorem is referenced by:  gacan  19232