|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > cmncom | Structured version Visualization version GIF version | ||
| Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| ablcom.b | ⊢ 𝐵 = (Base‘𝐺) | 
| ablcom.p | ⊢ + = (+g‘𝐺) | 
| Ref | Expression | 
|---|---|
| cmncom | ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ablcom.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | ablcom.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
| 3 | 1, 2 | iscmn 19808 | . . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) | 
| 4 | 3 | simprbi 496 | . . . 4 ⊢ (𝐺 ∈ CMnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) | 
| 5 | rsp2 3276 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))) | |
| 6 | 5 | imp 406 | . . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | 
| 7 | 4, 6 | sylan 580 | . . 3 ⊢ ((𝐺 ∈ CMnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) | 
| 8 | 7 | caovcomg 7629 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | 
| 9 | 8 | 3impb 1114 | 1 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | 
| 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-12 2176 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: ablcom 19818 cmn32 19819 cmn4 19820 cmn12 19821 cmnbascntr 19824 rinvmod 19825 mulgnn0di 19844 ghmcmn 19850 subcmn 19856 cntzcmn 19859 prdscmnd 19880 srgcom 20204 srgbinomlem4 20227 csrgbinom 20230 crngcom 20249 ip2di 21660 lply1binom 22315 chfacfscmulgsum 22867 chfacfpmmulgsum 22871 cpmadugsumlemF 22883 cmn246135 33039 cmn145236 33040 gsumwun 33069 omndadd2d 33086 rlocaddval 33273 ofldchr 33345 elrspunsn 33458 evl1deg2 33603 evl1deg3 33604 primrootsunit1 42099 aks6d1c5lem3 42139 | 
| Copyright terms: Public domain | W3C validator |