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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 5204 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | jctr 527 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V)) |
3 | f0 6555 | . . . 4 ⊢ ∅:∅⟶∅ | |
4 | xp0 6010 | . . . . 5 ⊢ ((Base‘𝐺) × ∅) = ∅ | |
5 | 4 | feq2i 6501 | . . . 4 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅) |
6 | 3, 5 | mpbir 233 | . . 3 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅ |
7 | ral0 4456 | . . 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 473 | . 2 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥)))) |
9 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
10 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
11 | eqid 2821 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
12 | 9, 10, 11 | isga 18415 | . 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 592 | 1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ∅c0 4291 × cxp 5548 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 0gc0g 16707 Grpcgrp 18097 GrpAct cga 18413 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-ga 18414 |
This theorem is referenced by: (None) |
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