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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 19822 | . . . . 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 3275 | . . . . 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 7628 | . 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 1537 ∈ wcel 2106 ∀wral 3059 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 Mndcmnd 18760 CMndccmn 19813 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-cmn 19815 |
This theorem is referenced by: ablcom 19832 cmn32 19833 cmn4 19834 cmn12 19835 cmnbascntr 19838 rinvmod 19839 mulgnn0di 19858 ghmcmn 19864 subcmn 19870 cntzcmn 19873 prdscmnd 19894 srgcom 20224 srgbinomlem4 20247 csrgbinom 20250 crngcom 20269 ip2di 21677 lply1binom 22330 chfacfscmulgsum 22882 chfacfpmmulgsum 22886 cpmadugsumlemF 22898 cmn246135 33021 cmn145236 33022 gsumwun 33051 omndadd2d 33068 rlocaddval 33255 ofldchr 33324 elrspunsn 33437 evl1deg2 33582 evl1deg3 33583 primrootsunit1 42079 aks6d1c5lem3 42119 |
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