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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 18022 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
2 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | cmncom 18032 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
5 | 1, 4 | syl3an1 1351 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 CMndccmn 18016 Abelcabl 18017 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-cmn 18018 df-abl 18019 |
This theorem is referenced by: ablinvadd 18038 ablsub2inv 18039 ablsubadd 18040 abladdsub 18043 ablpncan3 18045 ablsub32 18050 ablnnncan 18051 ablsubsub23 18053 eqgabl 18063 subgabl 18064 ablnsg 18073 lsmcomx 18082 qusabl 18091 frgpnabl 18101 ngplcan 22225 clmnegsubdi2 22713 clmvsubval2 22718 ncvspi 22764 r1pid 23723 abliso 29027 cnaddcom 33277 toycom 33278 lflsub 33372 lfladdcom 33377 |
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