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Mirrors > Home > MPE Home > Th. List > ismgmd | Structured version Visualization version GIF version |
Description: Deduce a magma from its properties. (Contributed by AV, 25-Feb-2020.) |
Ref | Expression |
---|---|
ismgmd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
ismgmd.0 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
ismgmd.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
ismgmd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
Ref | Expression |
---|---|
ismgmd | ⊢ (𝜑 → 𝐺 ∈ Mgm) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ismgmd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) | |
2 | 1 | 3expb 1118 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵) |
3 | 2 | ralrimivva 3197 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵) |
4 | ismgmd.b | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
5 | ismgmd.p | . . . . . . 7 ⊢ (𝜑 → + = (+g‘𝐺)) | |
6 | 5 | oveqd 7437 | . . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
7 | 6, 4 | eleq12d 2823 | . . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
8 | 4, 7 | raleqbidv 3339 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
9 | 4, 8 | raleqbidv 3339 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
10 | 3, 9 | mpbid 231 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
11 | ismgmd.0 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
12 | eqid 2728 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
13 | eqid 2728 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
14 | 12, 13 | ismgm 18601 | . . 3 ⊢ (𝐺 ∈ 𝑉 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
15 | 11, 14 | syl 17 | . 2 ⊢ (𝜑 → (𝐺 ∈ Mgm ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
16 | 10, 15 | mpbird 257 | 1 ⊢ (𝜑 → 𝐺 ∈ Mgm) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∀wral 3058 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 Mgmcmgm 18598 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-mgm 18600 |
This theorem is referenced by: issubmgm2 18663 |
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