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Theorem ga0 17952
 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 4943 . . 3 ∅ ∈ V
21jctr 566 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3 f0 6248 . . . . 5 ∅:∅⟶∅
4 xp0 5711 . . . . . 6 ((Base‘𝐺) × ∅) = ∅
54feq2i 6199 . . . . 5 (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
63, 5mpbir 221 . . . 4 ∅:((Base‘𝐺) × ∅)⟶∅
7 ral0 4221 . . . 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 2761 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2761 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2761 . . 3 (0g𝐺) = (0g𝐺)
1310, 11, 12isga 17945 . 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 701 1 (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2140  ∀wral 3051  Vcvv 3341  ∅c0 4059   × cxp 5265  ⟶wf 6046  ‘cfv 6050  (class class class)co 6815  Basecbs 16080  +gcplusg 16164  0gc0g 16323  Grpcgrp 17644   GrpAct cga 17943 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-map 8028  df-ga 17944 This theorem is referenced by: (None)
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