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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 12676 | . . . . . 6 ⊢ Base Fn V | |
2 | fnrel 5352 | . . . . . 6 ⊢ (Base Fn V → Rel Base) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Rel Base |
4 | relelfvdm 5586 | . . . . 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 2288 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ dom Base) |
8 | 7 | elexd 2773 | . 2 ⊢ (𝐴 ∈ 𝐵 → 𝑀 ∈ V) |
9 | ismgmn0.o | . . 3 ⊢ ⚬ = (+g‘𝑀) | |
10 | 6, 9 | ismgm 12940 | . 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 1364 ∈ wcel 2164 ∀wral 2472 Vcvv 2760 dom cdm 4659 Rel wrel 4664 Fn wfn 5249 ‘cfv 5254 (class class class)co 5918 Basecbs 12618 +gcplusg 12695 Mgmcmgm 12937 |
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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-cnex 7963 ax-resscn 7964 ax-1re 7966 ax-addrcl 7969 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-v 2762 df-sbc 2986 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-iota 5215 df-fun 5256 df-fn 5257 df-fv 5262 df-ov 5921 df-inn 8983 df-2 9041 df-ndx 12621 df-slot 12622 df-base 12624 df-plusg 12708 df-mgm 12939 |
This theorem is referenced by: mgm1 12953 opifismgmdc 12954 issgrpn0 12988 |
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