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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 3196 | . . 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 7367 | . . . . . 6 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
7 | 6, 4 | eleq12d 2833 | . . . . 5 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 ↔ (𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
8 | 4, 7 | raleqbidv 3318 | . . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ∈ 𝐵 ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) ∈ (Base‘𝐺))) |
9 | 4, 8 | raleqbidv 3318 | . . 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 2738 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
13 | eqid 2738 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
14 | 12, 13 | ismgm 18434 | . . 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 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3063 ‘cfv 6492 (class class class)co 7350 Basecbs 17019 +gcplusg 17069 Mgmcmgm 18431 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-nul 5262 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6444 df-fv 6500 df-ov 7353 df-mgm 18433 |
This theorem is referenced by: issubmgm2 45875 |
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