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Theorem gagrpid 17934
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 2771 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2771 . . . . . 6 (+g𝐺) = (+g𝐺)
3 gagrpid.1 . . . . . 6 0 = (0g𝐺)
41, 2, 3isga 17931 . . . . 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 484 . . . 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 483 . . 3 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))))
7 simpl 468 . . . 4 ((( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ( 0 𝑥) = 𝑥)
87ralimi 3101 . . 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 6804 . . . 4 (𝑥 = 𝐴 → ( 0 𝑥) = ( 0 𝐴))
11 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
1210, 11eqeq12d 2786 . . 3 (𝑥 = 𝐴 → (( 0 𝑥) = 𝑥 ↔ ( 0 𝐴) = 𝐴))
1312rspccva 3459 . 2 ((∀𝑥𝑌 ( 0 𝑥) = 𝑥𝐴𝑌) → ( 0 𝐴) = 𝐴)
149, 13sylan 569 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351   × cxp 5248  wf 6026  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Grpcgrp 17630   GrpAct cga 17929
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-map 8015  df-ga 17930
This theorem is referenced by:  gafo  17936  gass  17941  gasubg  17942  galcan  17944  gacan  17945  gaorber  17948  gastacl  17949
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