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Mirrors > Home > MPE Home > Th. List > ga0 | Structured version Visualization version GIF version |
Description: The action of a group on the empty set. (Contributed by Jeff Hankins, 11-Aug-2009.) (Revised by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
ga0 | ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5307 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | jctr 526 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V)) |
3 | f0 6770 | . . . 4 ⊢ ∅:∅⟶∅ | |
4 | xp0 6155 | . . . . 5 ⊢ ((Base‘𝐺) × ∅) = ∅ | |
5 | 4 | feq2i 6707 | . . . 4 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅) |
6 | 3, 5 | mpbir 230 | . . 3 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅ |
7 | ral0 4512 | . . 3 ⊢ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))) | |
8 | 6, 7 | pm3.2i 472 | . 2 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥)))) |
9 | eqid 2733 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
10 | eqid 2733 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | eqid 2733 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 9, 10, 11 | isga 19150 | . 2 ⊢ (∅ ∈ (𝐺 GrpAct ∅) ↔ ((𝐺 ∈ Grp ∧ ∅ ∈ V) ∧ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥)))))) |
13 | 2, 8, 12 | sylanblrc 591 | 1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ∅c0 4322 × cxp 5674 ⟶wf 6537 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 +gcplusg 17194 0gc0g 17382 Grpcgrp 18816 GrpAct cga 19148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-fv 6549 df-ov 7409 df-oprab 7410 df-mpo 7411 df-map 8819 df-ga 19149 |
This theorem is referenced by: (None) |
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