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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 4456 | . 2 ⊢ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅ | |
2 | 0mgm.b | . . . 4 ⊢ (Base‘𝑀) = ∅ | |
3 | 2 | eqcomi 2830 | . . 3 ⊢ ∅ = (Base‘𝑀) |
4 | eqid 2821 | . . 3 ⊢ (+g‘𝑀) = (+g‘𝑀) | |
5 | 3, 4 | ismgm 17853 | . 2 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ ∅ ∀𝑦 ∈ ∅ (𝑥(+g‘𝑀)𝑦) ∈ ∅)) |
6 | 1, 5 | mpbiri 260 | 1 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 Mgmcmgm 17850 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-mgm 17852 |
This theorem is referenced by: (None) |
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