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Mirrors > Home > ILE Home > Th. List > iscmnd | GIF version |
Description: Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
iscmnd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
iscmnd.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
iscmnd.g | ⊢ (𝜑 → 𝐺 ∈ Mnd) |
iscmnd.c | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
Ref | Expression |
---|---|
iscmnd | ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iscmnd.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd) | |
2 | iscmnd.c | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | |
3 | 2 | 3expib 1206 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
4 | 3 | ralrimivv 2558 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
5 | iscmnd.b | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
6 | iscmnd.p | . . . . . . . 8 ⊢ (𝜑 → + = (+g‘𝐺)) | |
7 | 6 | oveqd 5894 | . . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
8 | 6 | oveqd 5894 | . . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥)) |
9 | 7, 8 | eqeq12d 2192 | . . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
10 | 5, 9 | raleqbidv 2685 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
11 | 5, 10 | raleqbidv 2685 | . . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
12 | 11 | anbi2d 464 | . . 3 ⊢ (𝜑 → ((𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥)))) |
13 | 1, 4, 12 | mpbi2and 943 | . 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
14 | eqid 2177 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2177 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 14, 15 | iscmn 13101 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
17 | 13, 16 | sylibr 134 | 1 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ‘cfv 5218 (class class class)co 5877 Basecbs 12464 +gcplusg 12538 Mndcmnd 12822 CMndccmn 13093 |
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-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-un 3135 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-iota 5180 df-fv 5226 df-ov 5880 df-cmn 13095 |
This theorem is referenced by: isabld 13107 subcmnd 13134 iscrngd 13226 |
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