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| Mirrors > Home > MPE Home > Th. List > iscmnd | Structured version Visualization version 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 1122 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
| 4 | 3 | ralrimivv 3199 | . . 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 7449 | . . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
| 8 | 6 | oveqd 7449 | . . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥)) |
| 9 | 7, 8 | eqeq12d 2752 | . . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
| 10 | 5, 9 | raleqbidv 3345 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
| 11 | 5, 10 | raleqbidv 3345 | . . . 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 630 | . . 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 712 | . 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
| 14 | eqid 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 15 | eqid 2736 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 16 | 14, 15 | iscmn 19808 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
| 17 | 13, 16 | sylibr 234 | 1 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 +gcplusg 17298 Mndcmnd 18748 CMndccmn 19799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-cmn 19801 |
| This theorem is referenced by: isabld 19814 subcmn 19856 cntrcmnd 19861 prdscmnd 19880 iscrngd 20290 xrsmcmn 21405 psrcrng 21993 0ringcring 33257 idlsrgcmnd 33544 primrootsunit1 42099 2zrngacmnd 48169 |
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