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Theorem ga0 18373
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 5208 . . 3 ∅ ∈ V
21jctr 525 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3 f0 6559 . . . 4 ∅:∅⟶∅
4 xp0 6014 . . . . 5 ((Base‘𝐺) × ∅) = ∅
54feq2i 6505 . . . 4 (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
63, 5mpbir 232 . . 3 ∅:((Base‘𝐺) × ∅)⟶∅
7 ral0 4459 . . 3 𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥)))
86, 7pm3.2i 471 . 2 (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥))))
9 eqid 2826 . . 3 (Base‘𝐺) = (Base‘𝐺)
10 eqid 2826 . . 3 (+g𝐺) = (+g𝐺)
11 eqid 2826 . . 3 (0g𝐺) = (0g𝐺)
129, 10, 11isga 18366 . 2 (∅ ∈ (𝐺 GrpAct ∅) ↔ ((𝐺 ∈ Grp ∧ ∅ ∈ V) ∧ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥))))))
132, 8, 12sylanblrc 590 1 (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  wral 3143  Vcvv 3500  c0 4295   × cxp 5552  wf 6350  cfv 6354  (class class class)co 7150  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Grpcgrp 18048   GrpAct cga 18364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-map 8403  df-ga 18365
This theorem is referenced by: (None)
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