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| Mirrors > Home > MPE Home > Th. List > ablcom | Structured version Visualization version GIF version | ||
| Description: An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.) |
| Ref | Expression |
|---|---|
| ablcom.b | ⊢ 𝐵 = (Base‘𝐺) |
| ablcom.p | ⊢ + = (+g‘𝐺) |
| Ref | Expression |
|---|---|
| ablcom | ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ablcmn 19805 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
| 2 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 3 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
| 4 | 2, 3 | cmncom 19816 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| 5 | 1, 4 | syl3an1 1164 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 CMndccmn 19798 Abelcabl 19799 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-cmn 19800 df-abl 19801 |
| This theorem is referenced by: ablinvadd 19825 ablsub2inv 19826 ablsubadd 19827 abladdsub 19830 ablsubaddsub 19832 ablpncan3 19834 ablsub32 19839 ablnnncan 19840 ablsubsub23 19842 eqgabl 19852 subgabl 19854 ablnsg 19865 lsmcomx 19874 qusabl 19883 frgpnabl 19893 imasabl 19894 subrngringnsg 20553 ngplcan 24624 clmnegsubdi2 25138 clmvsubval2 25143 ncvspi 25190 r1pid 26200 abliso 33041 r1plmhm 33630 lindsunlem 33675 cnaddcom 38973 toycom 38974 lflsub 39068 lfladdcom 39073 |
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