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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 5175 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | jctr 528 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V)) |
3 | f0 6534 | . . . 4 ⊢ ∅:∅⟶∅ | |
4 | xp0 5982 | . . . . 5 ⊢ ((Base‘𝐺) × ∅) = ∅ | |
5 | 4 | feq2i 6479 | . . . 4 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅) |
6 | 3, 5 | mpbir 234 | . . 3 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅ |
7 | ral0 4414 | . . 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 474 | . 2 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥)))) |
9 | eqid 2798 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
10 | eqid 2798 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | eqid 2798 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 9, 10, 11 | isga 18413 | . 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 593 | 1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∅c0 4243 × cxp 5517 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 0gc0g 16705 Grpcgrp 18095 GrpAct cga 18411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-ga 18412 |
This theorem is referenced by: (None) |
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