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Theorem gagrpid 17550
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 = (0g𝐺)
Assertion
Ref Expression
gagrpid (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)

Proof of Theorem gagrpid
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2610 . . . . . 6 (+g𝐺) = (+g𝐺)
3 gagrpid.1 . . . . . 6 0 = (0g𝐺)
41, 2, 3isga 17547 . . . . 5 ( ∈ (𝐺 GrpAct 𝑌) ↔ ((𝐺 ∈ Grp ∧ 𝑌 ∈ V) ∧ ( :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))))
54simprbi 479 . . . 4 ( ∈ (𝐺 GrpAct 𝑌) → ( :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
65simprd 478 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
7 simpl 472 . . . 4 ((( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ( 0 𝑥) = 𝑥)
87ralimi 2936 . . 3 (∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌 ( 0 𝑥) = 𝑥)
96, 8syl 17 . 2 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 ( 0 𝑥) = 𝑥)
10 oveq2 6557 . . . 4 (𝑥 = 𝐴 → ( 0 𝑥) = ( 0 𝐴))
11 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2625 . . 3 (𝑥 = 𝐴 → (( 0 𝑥) = 𝑥 ↔ ( 0 𝐴) = 𝐴))
1312rspccva 3281 . 2 ((∀𝑥𝑌 ( 0 𝑥) = 𝑥𝐴𝑌) → ( 0 𝐴) = 𝐴)
149, 13sylan 487 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173   × cxp 5036  wf 5800  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245   GrpAct cga 17545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-ga 17546
This theorem is referenced by:  gafo  17552  gass  17557  gasubg  17558  galcan  17560  gacan  17561  gaorber  17564  gastacl  17565
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