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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0mgm | Structured version Visualization version GIF version |
Description: A set with an empty base set is always a magma. (Contributed by AV, 25-Feb-2020.) |
Ref | Expression |
---|---|
0mgm.b | ⊢ (Base‘𝑀) = ∅ |
Ref | Expression |
---|---|
0mgm | ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ral0 4414 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅ | |
2 | 0mgm.b | . . . 4 ⊢ (Base‘𝑀) = ∅ | |
3 | 2 | eqcomi 2807 | . . 3 ⊢ ∅ = (Base‘𝑀) |
4 | eqid 2798 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
5 | 3, 4 | ismgm 17845 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅)) |
6 | 1, 5 | mpbiri 261 | 1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∅c0 4243 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 Mgmcmgm 17842 |
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-nul 5174 |
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-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-mgm 17844 |
This theorem is referenced by: (None) |
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