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Theorem gagrpid 18542
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 18539 . . . 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 500 . . 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 486 . . . 4 ((( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ( 0 𝑥) = 𝑥)
76ralimi 3075 . . 3 (∀𝑥𝑌 (( 0 𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧) 𝑥) = (𝑦 (𝑧 𝑥))) → ∀𝑥𝑌 ( 0 𝑥) = 𝑥)
85, 7simpl2im 507 . 2 ( ∈ (𝐺 GrpAct 𝑌) → ∀𝑥𝑌 ( 0 𝑥) = 𝑥)
9 oveq2 7178 . . . 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 3525 . 2 ((∀𝑥𝑌 ( 0 𝑥) = 𝑥𝐴𝑌) → ( 0 𝐴) = 𝐴)
138, 12sylan 583 1 (( ∈ (𝐺 GrpAct 𝑌) ∧ 𝐴𝑌) → ( 0 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  wral 3053  Vcvv 3398   × cxp 5523  wf 6335  cfv 6339  (class class class)co 7170  Basecbs 16586  +gcplusg 16668  0gc0g 16816  Grpcgrp 18219   GrpAct cga 18537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-map 8439  df-ga 18538
This theorem is referenced by:  gafo  18544  gass  18549  gasubg  18550  galcan  18552  gacan  18553  gaorber  18556  gastacl  18557
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