Theorem gagrpid 19559

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 2763	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2763	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		gagrpid.1	. . . . 5 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	isga 19556	. . . 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 503	. . 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 488	. . . 4 ⊢ ((( 0 ⊕ 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧) ⊕ 𝑥) = (𝑦 ⊕ (𝑧 ⊕ 𝑥))) → ( 0 ⊕ 𝑥) = 𝑥)
7	6	ralimi 3103	. . 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 513	. 2 ⊢ ( ⊕ ∈ (𝐺 GrpAct 𝑌) → ∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥)
9		oveq2 7459	. . . 4 ⊢ (𝑥 = 𝐴 → ( 0 ⊕ 𝑥) = ( 0 ⊕ 𝐴))
10		id 23	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2779	. . 3 ⊢ (𝑥 = 𝐴 → (( 0 ⊕ 𝑥) = 𝑥 ↔ ( 0 ⊕ 𝐴) = 𝐴))
12	11	rspccva 3579	. 2 ⊢ ((∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)
13	8, 12	sylan 592	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   × cxp 5656  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  0_gc0g 17660  Grpcgrp 19194   GrpAct cga 19554

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-ga 19555

This theorem is used by:  gafo  19561  gass  19566  gasubg  19567  galcan  19569  gacan  19570  gaorber  19573  gastacl  19574