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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 4718 | . . 3 ⊢ ∅ ∈ V | |
2 | 1 | jctr 563 | . 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Grp ∧ ∅ ∈ V)) |
3 | f0 5999 | . . . . 5 ⊢ ∅:∅⟶∅ | |
4 | xp0 5471 | . . . . . 6 ⊢ ((Base‘𝐺) × ∅) = ∅ | |
5 | 4 | feq2i 5950 | . . . . 5 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ↔ ∅:∅⟶∅) |
6 | 3, 5 | mpbir 220 | . . . 4 ⊢ ∅:((Base‘𝐺) × ∅)⟶∅ |
7 | ral0 4028 | . . . 4 ⊢ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))) | |
8 | 6, 7 | pm3.2i 470 | . . 3 ⊢ (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥)))) |
9 | 8 | a1i 11 | . 2 ⊢ (𝐺 ∈ Grp → (∅:((Base‘𝐺) × ∅)⟶∅ ∧ ∀𝑥 ∈ ∅ (((0g‘𝐺)∅𝑥) = 𝑥 ∧ ∀𝑦 ∈ (Base‘𝐺)∀𝑧 ∈ (Base‘𝐺)((𝑦(+g‘𝐺)𝑧)∅𝑥) = (𝑦∅(𝑧∅𝑥))))) |
10 | eqid 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
11 | eqid 2610 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
12 | eqid 2610 | . . 3 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
13 | 10, 11, 12 | isga 17547 | . 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)))))) |
14 | 2, 9, 13 | sylanbrc 695 | 1 ⊢ (𝐺 ∈ Grp → ∅ ∈ (𝐺 GrpAct ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ∅c0 3874 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 Grpcgrp 17245 GrpAct cga 17545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-map 7746 df-ga 17546 |
This theorem is referenced by: (None) |
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