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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 4514 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅ | |
2 | 0mgm.b | . . . 4 ⊢ (Base‘𝑀) = ∅ | |
3 | 2 | eqcomi 2734 | . . 3 ⊢ ∅ = (Base‘𝑀) |
4 | eqid 2725 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
5 | 3, 4 | ismgm 18604 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅)) |
6 | 1, 5 | mpbiri 257 | 1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∅c0 4322 ‘cfv 6549 (class class class)co 7419 Basecbs 17183 +gcplusg 17236 Mgmcmgm 18601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2930 df-ral 3051 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-iota 6501 df-fv 6557 df-ov 7422 df-mgm 18603 |
This theorem is referenced by: (None) |
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