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Theorem ga0 17554
Description: The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.)
Assertion
Ref Expression
ga0 (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))

Proof of Theorem ga0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4718 . . 3 ∅ ∈ V
21jctr 563 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3 f0 5999 . . . . 5 ∅:∅⟶∅
4 xp0 5471 . . . . . 6 ((Base‘𝐺) × ∅) = ∅
54feq2i 5950 . . . . 5 (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
63, 5mpbir 220 . . . 4 ∅:((Base‘𝐺) × ∅)⟶∅
7 ral0 4028 . . . 4 𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥)))
86, 7pm3.2i 470 . . 3 (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥))))
98a1i 11 . 2 (𝐺 ∈ Grp → (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥)))))
10 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2610 . . 3 (0g𝐺) = (0g𝐺)
1310, 11, 12isga 17547 . 2 (∅ ∈ (𝐺 GrpAct ∅) ↔ ((𝐺 ∈ Grp ∧ ∅ ∈ V) ∧ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥))))))
142, 9, 13sylanbrc 695 1 (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  c0 3874   × 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: (None)
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