Theorem galcan 19162

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 7413	. . 3 ⊢ ((𝐴 ⊕ 𝐵) = (𝐴 ⊕ 𝐶) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)))
2		simpl 483	. . . . . . . 8 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ⊕ ∈ (𝐺 GrpAct 𝑌))
3		gagrp 19150	. . . . . . . 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 1194	. . . . . . 7 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐴 ∈ 𝑋)
6		galcan.1	. . . . . . . 8 ⊢ 𝑋 = (Base‘𝐺)
7		eqid 2732	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2732	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
9		eqid 2732	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	6, 7, 8, 9	grplinv 18870	. . . . . . 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 7420	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = ((0g‘𝐺) ⊕ 𝐵))
13	6, 9	grpinvcl 18868	. . . . . . 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 1195	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐵 ∈ 𝑌)
16	6, 7	gaass 19155	. . . . . 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 1372	. . . . 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 19152	. . . . . 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 2780	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = 𝐵)
21	11	oveq1d 7420	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐶) = ((0g‘𝐺) ⊕ 𝐶))
22		simpr3 1196	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → 𝐶 ∈ 𝑌)
23	6, 7	gaass 19155	. . . . . 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 1372	. . . . 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 19152	. . . . . 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 2780	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) = 𝐶)
28	20, 27	eqeq12d 2748	. . 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 7413	. 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381  Grpcgrp 18815  inv_gcminusg 18816   GrpAct cga 19147

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-ga 19148

This theorem is referenced by:  gacan  19163