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| Mirrors > Home > MPE Home > Th. List > ismgmn0 | Structured version Visualization version GIF version | ||
| Description: The predicate "is a magma" for a structure with a nonempty base set. (Contributed by AV, 29-Jan-2020.) |
| Ref | Expression |
|---|---|
| ismgmn0.b | ⊢ 𝐵 = (Base‘𝑀) |
| ismgmn0.o | ⊢ ⚬ = (+g‘𝑀) |
| Ref | Expression |
|---|---|
| ismgmn0 | ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ismgmn0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
| 2 | 1 | eleq2i 2833 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑀)) |
| 3 | 2 | biimpi 216 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑀)) |
| 4 | 3 | elfvexd 6945 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V) |
| 5 | ismgmn0.o | . . 3 ⊢ ⚬ = (+g‘𝑀) | |
| 6 | 1, 5 | ismgm 18654 | . 2 ⊢ (𝑀 ∈ V → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
| 7 | 4, 6 | syl 17 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 |
| This theorem is referenced by: mgmpropd 18664 mgm1 18671 opifismgm 18672 issgrpn0 18735 xrsmgm 21419 opmpoismgm 48083 nnsgrpmgm 48092 2zrngamgm 48161 2zrngmmgm 48168 |
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