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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 1124 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
4 | 3 | ralrimivv 3101 | . . 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 7208 | . . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
8 | 6 | oveqd 7208 | . . . . . . 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 3303 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
11 | 5, 10 | raleqbidv 3303 | . . . 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 632 | . . 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 19132 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
17 | 13, 16 | sylibr 237 | 1 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 Mndcmnd 18127 CMndccmn 19124 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-cmn 19126 |
This theorem is referenced by: isabld 19138 subcmn 19176 cntrcmnd 19181 prdscmnd 19200 iscrngd 19558 xrsmcmn 20340 psrcrng 20892 idlsrgcmnd 31328 2zrngacmnd 45116 |
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