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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 1260 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
4 | 3 | ralrimivv 2953 | . . 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 6566 | . . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
8 | 6 | oveqd 6566 | . . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥)) |
9 | 7, 8 | eqeq12d 2625 | . . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
10 | 5, 9 | raleqbidv 3129 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
11 | 5, 10 | raleqbidv 3129 | . . . 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 736 | . . 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 958 | . 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
14 | eqid 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2610 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 14, 15 | iscmn 18023 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
17 | 13, 16 | sylibr 223 | 1 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Mndcmnd 17117 CMndccmn 18016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-cmn 18018 |
This theorem is referenced by: isabld 18029 subcmn 18065 prdscmnd 18087 iscrngd 18409 psrcrng 19234 xrsmcmn 19588 2zrngacmnd 41732 |
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