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Theorem gagrpid 18362
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 2818 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2818 . . . . 5 (+g𝐺) = (+g𝐺)
3 gagrpid.1 . . . . 5 0 = (0g𝐺)
41, 2, 3isga 18359 . . . 4 ( ∈ (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 497 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → ( :((Base‘𝐺) × 𝑌)⟶𝑌 ∧ ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥)))))
6 simpl 483 . . . 4 ((( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ( 0 𝑥) = 𝑥)
76ralimi 3157 . . 3 (∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌 ( 0 𝑥) = 𝑥)
85, 7simpl2im 504 . 2 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 ( 0 𝑥) = 𝑥)
9 oveq2 7153 . . . 4 (𝑥 = 𝐴 → ( 0 𝑥) = ( 0 𝐴))
10 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
119, 10eqeq12d 2834 . . 3 (𝑥 = 𝐴 → (( 0 𝑥) = 𝑥 ↔ ( 0 𝐴) = 𝐴))
1211rspccva 3619 . 2 ((∀𝑥𝑌 ( 0 𝑥) = 𝑥𝐴𝑌) → ( 0 𝐴) = 𝐴)
138, 12sylan 580 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492   × cxp 5546  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  0gc0g 16701  Grpcgrp 18041   GrpAct cga 18357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-ga 18358
This theorem is referenced by:  gafo  18364  gass  18369  gasubg  18370  galcan  18372  gacan  18373  gaorber  18376  gastacl  18377
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