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Mirrors > Home > ILE Home > Th. List > ismgmn0 | 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 | basfn 12489 | . . . . . 6 ⊢ Base Fn V | |
2 | fnrel 5309 | . . . . . 6 ⊢ (Base Fn V → Rel Base) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Rel Base |
4 | relelfvdm 5542 | . . . . 5 ⊢ ((Rel Base ∧ 𝐴 ∈ (Base‘𝑀)) → 𝑀 ∈ dom Base) | |
5 | 3, 4 | mpan 424 | . . . 4 ⊢ (𝐴 ∈ (Base‘𝑀) → 𝑀 ∈ dom Base) |
6 | ismgmn0.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
7 | 5, 6 | eleq2s 2272 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ dom Base) |
8 | 7 | elexd 2750 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V) |
9 | ismgmn0.o | . . 3 ⊢ ⚬ = (+g‘𝑀) | |
10 | 6, 9 | ismgm 12655 | . 2 ⊢ (𝑀 ∈ V → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
11 | 8, 10 | syl 14 | 1 ⊢ (𝐴 ∈ 𝐵 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ∀wral 2455 Vcvv 2737 dom cdm 4622 Rel wrel 4627 Fn wfn 5206 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 Mgmcmgm 12652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-mgm 12654 |
This theorem is referenced by: mgm1 12668 opifismgmdc 12669 issgrpn0 12690 |
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