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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 2898 | . . . 4 ⊢ (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ (Base‘𝑀)) |
3 | 2 | biimpi 208 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ (Base‘𝑀)) |
4 | 3 | elfvexd 6472 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V) |
5 | ismgmn0.o | . . 3 ⊢ ⚬ = (+g‘𝑀) | |
6 | 1, 5 | ismgm 17603 | . 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 198 = wceq 1656 ∈ wcel 2164 ∀wral 3117 Vcvv 3414 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 Mgmcmgm 17600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 ax-pow 5067 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-dm 5356 df-iota 6090 df-fv 6135 df-ov 6913 df-mgm 17602 |
This theorem is referenced by: mgm1 17617 opifismgm 17618 issgrpn0 17647 xrsmgm 20148 mgmpropd 42636 opmpt2ismgm 42668 nnsgrpmgm 42677 2zrngamgm 42800 2zrngmmgm 42807 |
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