Theorem gagrpid 19334

Description: The identity of the group does not alter the base set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)

Hypothesis

Ref	Expression
gagrpid.1	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
gagrpid	⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)

Proof of Theorem gagrpid

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2740	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2740	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		gagrpid.1	. . . . 5 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	isga 19331	. . . 4 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))))))
5	4	simprbi 496	. . 3 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ( ⊕ :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥)))))
6		simpl 482	. . . 4 ⊢ ((( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ( 0 ⊕ 𝑥) = 𝑥)
7	6	ralimi 3089	. . 3 ⊢ (∀𝑥 ∈ 𝑌 (( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥)
8	5, 7	simpl2im 503	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥)
9		oveq2 7456	. . . 4 ⊢ (𝑥 = 𝐴 → ( 0 ⊕ 𝑥) = ( 0 ⊕ 𝐴))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2756	. . 3 ⊢ (𝑥 = 𝐴 → (( 0 ⊕ 𝑥) = 𝑥 ↔ ( 0 ⊕ 𝐴) = 𝐴))
12	11	rspccva 3634	. 2 ⊢ ((∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)
13	8, 12	sylan 579	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   × cxp 5698  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  Basecbs 17258  +_gcplusg 17311  0_gc0g 17499  Grpcgrp 18973   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-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-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-ga 19330

This theorem is referenced by:  gafo  19336  gass  19341  gasubg  19342  galcan  19344  gacan  19345  gaorber  19348  gastacl  19349