Theorem galcan 19344

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 7456	. . 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 19332	. . . . . . . 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 2740	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2740	. . . . . . . 8 ⊢ (0g‘𝐺) = (0g‘𝐺)
9		eqid 2740	. . . . . . . 8 ⊢ (invg‘𝐺) = (invg‘𝐺)
10	6, 7, 8, 9	grplinv 19029	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺))
11	4, 5, 10	syl2anc 583	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) = (0g‘𝐺))
12	11	oveq1d 7463	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((((invg‘𝐺)‘𝐴)(+g‘𝐺)𝐴) ⊕ 𝐵) = ((0g‘𝐺) ⊕ 𝐵))
13	6, 9	grpinvcl 19027	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝐴 ∈ 𝑋) → ((invg‘𝐺)‘𝐴) ∈ 𝑋)
14	4, 5, 13	syl2anc 583	. . . . . 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 19337	. . . . . 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 19334	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐵 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵)
19	2, 15, 18	syl2anc 583	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐵) = 𝐵)
20	12, 17, 19	3eqtr3d 2788	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐵)) = 𝐵)
21	11	oveq1d 7463	. . . . 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 19337	. . . . . 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 19334	. . . . . 6 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐶 ∈ 𝑌) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
26	2, 22, 25	syl2anc 583	. . . . 5 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → ((0g‘𝐺) ⊕ 𝐶) = 𝐶)
27	21, 24, 26	3eqtr3d 2788	. . . 4 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑌 ∧ 𝐶 ∈ 𝑌)) → (((invg‘𝐺)‘𝐴) ⊕ (𝐴 ⊕ 𝐶)) = 𝐶)
28	20, 27	eqeq12d 2756	. . 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 7456	. 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 1087   = wceq 1537   ∈ wcel 2108  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973  inv_gcminusg 18974   GrpAct cga 19329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-0g 17501  df-mgm 18678  df-sgrp 18757  df-mnd 18773  df-grp 18976  df-minusg 18977  df-ga 19330

This theorem is referenced by:  gacan  19345