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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 12502 | . . . . 5 ⊢ Base Fn V | |
2 | vex 2740 | . . . . 5 ⊢ 𝑚 ∈ V | |
3 | funfvex 5528 | . . . . . 6 ⊢ ((Fun Base ∧ 𝑚 ∈ dom Base) → (Base‘𝑚) ∈ V) | |
4 | 3 | funfni 5312 | . . . . 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 5511 | . . . 4 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀)) | |
8 | ismgm.b | . . . 4 ⊢ 𝐵 = (Base‘𝑀) | |
9 | 7, 8 | eqtr4di 2228 | . . 3 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵) |
10 | plusgslid 12551 | . . . . . . 7 ⊢ (+g = Slot (+g‘ndx) ∧ (+g‘ndx) ∈ ℕ) | |
11 | 10 | slotex 12472 | . . . . . 6 ⊢ (𝑚 ∈ V → (+g‘𝑚) ∈ V) |
12 | 11 | elv 2741 | . . . . 5 ⊢ (+g‘𝑚) ∈ V |
13 | 12 | a1i 9 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) ∈ V) |
14 | fveq2 5511 | . . . . . 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 2228 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) → (+g‘𝑚) = ⚬ ) |
18 | simplr 528 | . . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → 𝑏 = 𝐵) | |
19 | oveq 5875 | . . . . . . . 8 ⊢ (𝑜 = ⚬ → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦)) | |
20 | 19 | adantl 277 | . . . . . . 7 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (𝑥𝑜𝑦) = (𝑥 ⚬ 𝑦)) |
21 | 20, 18 | eleq12d 2248 | . . . . . 6 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → ((𝑥𝑜𝑦) ∈ 𝑏 ↔ (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
22 | 18, 21 | raleqbidv 2684 | . . . . 5 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
23 | 18, 22 | raleqbidv 2684 | . . . 4 ⊢ (((𝑚 = 𝑀 ∧ 𝑏 = 𝐵) ∧ 𝑜 = ⚬ ) → (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ⚬ 𝑦) ∈ 𝐵)) |
24 | 13, 17, 23 | sbcied2 3000 | . . 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 3000 | . 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 12667 | . 2 ⊢ Mgm = {𝑚 ∣ [(Base‘𝑚) / 𝑏][(+g‘𝑚) / 𝑜]∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥𝑜𝑦) ∈ 𝑏} | |
27 | 25, 26 | elab2g 2884 | 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 2148 ∀wral 2455 Vcvv 2737 [wsbc 2962 Fn wfn 5207 ‘cfv 5212 (class class class)co 5869 Basecbs 12445 +gcplusg 12518 Mgmcmgm 12665 |
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 4206 ax-un 4430 ax-cnex 7893 ax-resscn 7894 ax-1re 7896 ax-addrcl 7899 |
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 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 df-ov 5872 df-inn 8909 df-2 8967 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-mgm 12667 |
This theorem is referenced by: ismgmn0 12669 mgmcl 12670 mgm0 12680 issgrpv 12702 |
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