MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ga0 Structured version   Visualization version   GIF version

Theorem ga0 17647
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 4755 . . 3 ∅ ∈ V
21jctr 564 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V))
3 f0 6045 . . . . 5 ∅:∅⟶∅
4 xp0 5515 . . . . . 6 ((Base‘𝐺) × ∅) = ∅
54feq2i 5996 . . . . 5 (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅)
63, 5mpbir 221 . . . 4 ∅:((Base‘𝐺) × ∅)⟶∅
7 ral0 4053 . . . 4 𝑥 ∈ ∅ (((0g𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧𝑥)))
86, 7pm3.2i 471 . . 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 2626 . . 3 (Base‘𝐺) = (Base‘𝐺)
11 eqid 2626 . . 3 (+g𝐺) = (+g𝐺)
12 eqid 2626 . . 3 (0g𝐺) = (0g𝐺)
1310, 11, 12isga 17640 . 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 697 1 (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1992  wral 2912  Vcvv 3191  c0 3896   × cxp 5077  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  +gcplusg 15857  0gc0g 16016  Grpcgrp 17338   GrpAct cga 17638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-ga 17639
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator