Theorem gagrpid 19233

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 2730	. . . . 5 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2730	. . . . 5 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		gagrpid.1	. . . . 5 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	isga 19230	. . . 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 3067	. . 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 7398	. . . 4 ⊢ (𝑥 = 𝐴 → ( 0 ⊕ 𝑥) = ( 0 ⊕ 𝐴))
10		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
11	9, 10	eqeq12d 2746	. . 3 ⊢ (𝑥 = 𝐴 → (( 0 ⊕ 𝑥) = 𝑥 ↔ ( 0 ⊕ 𝐴) = 𝐴))
12	11	rspccva 3590	. 2 ⊢ ((∀𝑥 ∈ 𝑌 ( 0 ⊕ 𝑥) = 𝑥 ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)
13	8, 12	sylan 580	1 ⊢ (( ⊕ ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴 ∈ 𝑌) → ( 0 ⊕ 𝐴) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   × cxp 5639  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  0_gc0g 17409  Grpcgrp 18872   GrpAct cga 19228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-ga 19229

This theorem is referenced by:  gafo  19235  gass  19240  gasubg  19241  galcan  19243  gacan  19244  gaorber  19247  gastacl  19248