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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 2906 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑀)) |
3 | 2 | biimpi 218 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑀)) |
4 | 3 | elfvexd 6706 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V) |
5 | ismgmn0.o | . . 3 ⊢ ⚬ = (+g‘𝑀) | |
6 | 1, 5 | ismgm 17855 | . 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 208 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 Mgmcmgm 17852 |
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 2795 ax-nul 5212 ax-pow 5268 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-dm 5567 df-iota 6316 df-fv 6365 df-ov 7161 df-mgm 17854 |
This theorem is referenced by: mgm1 17870 opifismgm 17871 issgrpn0 17906 xrsmgm 20582 mgmpropd 44049 opmpoismgm 44081 nnsgrpmgm 44090 2zrngamgm 44217 2zrngmmgm 44224 |
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