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Mirrors > Home > ILE Home > Th. List > ismgm | GIF version |
Description: The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 6-Jan-2020.) |
Ref | Expression |
---|---|
ismgm.b | ⊢ 𝐵 = (Base‘𝑀) |
ismgm.o | ⊢ ⚬ = (+g‘𝑀) |
Ref | Expression |
---|---|
ismgm | ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | basfn 12486 | . . . . 5 ⊢ Base Fn V | |
2 | vex 2738 | . . . . 5 ⊢ 𝑚 ∈ V | |
3 | funfvex 5524 | . . . . . 6 ⊢ ((Fun Base ∧ 𝑚 ∈ dom Base) → (Base‘𝑚) ∈ V) | |
4 | 3 | funfni 5308 | . . . . 5 ⊢ ((Base Fn V ∧ 𝑚 ∈ V) → (Base‘𝑚) ∈ V) |
5 | 1, 2, 4 | mp2an 426 | . . . 4 ⊢ (Base‘𝑚) ∈ V |
6 | 5 | a1i 9 | . . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) ∈ V) |
7 | fveq2 5507 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
8 | ismgm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
9 | 7, 8 | eqtr4di 2226 | . . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
10 | plusgslid 12526 | . . . . . . 7 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
11 | 10 | slotex 12456 | . . . . . 6 ⊢ (𝑚 ∈ V → (+g‘𝑚) ∈ V) |
12 | 11 | elv 2739 | . . . . 5 ⊢ (+g‘𝑚) ∈ V |
13 | 12 | a1i 9 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) ∈ V) |
14 | fveq2 5507 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀)) | |
15 | 14 | adantr 276 | . . . . 5 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = (+g‘𝑀)) |
16 | ismgm.o | . . . . 5 ⊢ ⚬ = (+g‘𝑀) | |
17 | 15, 16 | eqtr4di 2226 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = ⚬ ) |
18 | simplr 528 | . . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵) | |
19 | oveq 5871 | . . . . . . . 8 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦)) | |
20 | 19 | adantl 277 | . . . . . . 7 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦)) |
21 | 20, 18 | eleq12d 2246 | . . . . . 6 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
22 | 18, 21 | raleqbidv 2682 | . . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
23 | 18, 22 | raleqbidv 2682 | . . . 4 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
24 | 13, 17, 23 | sbcied2 2998 | . . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → ([(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
25 | 6, 9, 24 | sbcied2 2998 | . 2 ⊢ (𝑚 = 𝑀 → ([(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
26 | df-mgm 12641 | . 2 ⊢ Mgm = {𝑚 ∣ [(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏} | |
27 | 25, 26 | elab2g 2882 | 1 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2146 ∀wral 2453 Vcvv 2735 [wsbc 2960 Fn wfn 5203 ‘cfv 5208 (class class class)co 5865 Basecbs 12429 +gcplusg 12493 Mgmcmgm 12639 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-cnex 7877 ax-resscn 7878 ax-1re 7880 ax-addrcl 7883 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ral 2458 df-rex 2459 df-v 2737 df-sbc 2961 df-un 3131 df-in 3133 df-ss 3140 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-iota 5170 df-fun 5210 df-fn 5211 df-fv 5216 df-ov 5868 df-inn 8893 df-2 8951 df-ndx 12432 df-slot 12433 df-base 12435 df-plusg 12506 df-mgm 12641 |
This theorem is referenced by: ismgmn0 12643 mgmcl 12644 mgm0 12654 issgrpv 12676 |
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