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Theorem gagrpid 18900
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 2738 . . . . 5 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2738 . . . . 5 (+g𝐺) = (+g𝐺)
3 gagrpid.1 . . . . 5 0 = (0g𝐺)
41, 2, 3isga 18897 . . . 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 3087 . . 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 7283 . . . 4 (𝑥 = 𝐴 → ( 0 𝑥) = ( 0 𝐴))
10 id 22 . . . 4 (𝑥 = 𝐴𝑥 = 𝐴)
119, 10eqeq12d 2754 . . 3 (𝑥 = 𝐴 → (( 0 𝑥) = 𝑥 ↔ ( 0 𝐴) = 𝐴))
1211rspccva 3560 . 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 1539  wcel 2106  wral 3064  Vcvv 3432   × cxp 5587  wf 6429  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Grpcgrp 18577   GrpAct cga 18895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-ga 18896
This theorem is referenced by:  gafo  18902  gass  18907  gasubg  18908  galcan  18910  gacan  18911  gaorber  18914  gastacl  18915
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