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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 2842 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑀)) |
3 | 2 | biimpi 219 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑀)) |
4 | 3 | elfvexd 6685 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V) |
5 | ismgmn0.o | . . 3 ⊢ ⚬ = (+g‘𝑀) | |
6 | 1, 5 | ismgm 17904 | . 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 209 = wceq 1539 ∈ wcel 2112 ∀wral 3068 Vcvv 3407 ‘cfv 6328 (class class class)co 7143 Basecbs 16526 +gcplusg 16608 Mgmcmgm 17901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-sep 5162 ax-nul 5169 ax-pr 5291 |
This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2899 df-ne 2950 df-ral 3073 df-rex 3074 df-v 3409 df-sbc 3694 df-dif 3857 df-un 3859 df-in 3861 df-ss 3871 df-nul 4222 df-if 4414 df-sn 4516 df-pr 4518 df-op 4522 df-uni 4792 df-br 5026 df-dm 5527 df-iota 6287 df-fv 6336 df-ov 7146 df-mgm 17903 |
This theorem is referenced by: mgm1 17919 opifismgm 17920 issgrpn0 17955 xrsmgm 20186 mgmpropd 44747 opmpoismgm 44779 nnsgrpmgm 44788 2zrngamgm 44915 2zrngmmgm 44922 |
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