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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 1118 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
4 | 3 | ralrimivv 3190 | . . 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 7172 | . . . . . . 7 ⊢ (𝜑 → (𝑥 + 𝑦) = (𝑥(+g‘𝐺)𝑦)) |
8 | 6 | oveqd 7172 | . . . . . . 7 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥)) |
9 | 7, 8 | eqeq12d 2837 | . . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
10 | 5, 9 | raleqbidv 3401 | . . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
11 | 5, 10 | raleqbidv 3401 | . . . 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 710 | . 2 ⊢ (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
14 | eqid 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
15 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
16 | 14, 15 | iscmn 18913 | . 2 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g‘𝐺)𝑦) = (𝑦(+g‘𝐺)𝑥))) |
17 | 13, 16 | sylibr 236 | 1 ⊢ (𝜑 → 𝐺 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 Mndcmnd 17910 CMndccmn 18905 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-cmn 18907 |
This theorem is referenced by: isabld 18919 subcmn 18956 cntrcmnd 18961 prdscmnd 18980 iscrngd 19335 psrcrng 20192 xrsmcmn 20567 2zrngacmnd 44212 |
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