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Theorem ga0 17939
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 4995 . . 3 ∅ ∈ V
21jctr 516 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3 f0 6308 . . . . 5 ∅:∅⟶∅
4 xp0 5774 . . . . . 6 ((Base‘𝐺) × ∅) = ∅
54feq2i 6255 . . . . 5 (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
63, 5mpbir 222 . . . 4 ∅:((Base‘𝐺) × ∅)⟶∅
7 ral0 4282 . . . 4 𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥)))
86, 7pm3.2i 458 . . 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 2817 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2817 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2817 . . 3 (0g𝐺) = (0g𝐺)
1310, 11, 12isga 17932 . 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 574 1 (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2157  wral 3107  Vcvv 3402  c0 4127   × cxp 5320  wf 6104  cfv 6108  (class class class)co 6881  Basecbs 16075  +gcplusg 16160  0gc0g 16312  Grpcgrp 17634   GrpAct cga 17930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-id 5230  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-fv 6116  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-map 8101  df-ga 17931
This theorem is referenced by: (None)
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