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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 1121 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵) |
| 3 | 2 | ralrimivva 3202 | . . 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 7448 | . . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
| 7 | 6, 4 | eleq12d 2835 | . . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
| 8 | 4, 7 | raleqbidv 3346 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
| 9 | 4, 8 | raleqbidv 3346 | . . 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 232 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺)) |
| 11 | ismgmd.0 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
| 12 | eqid 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 13 | eqid 2737 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 14 | 12, 13 | ismgm 18654 | . . 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 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Mgmcmgm 18651 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 |
| This theorem is referenced by: issubmgm2 18716 |
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