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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 18913 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
2 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | cmncom 18923 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
5 | 1, 4 | syl3an1 1159 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 CMndccmn 18906 Abelcabl 18907 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-cmn 18908 df-abl 18909 |
This theorem is referenced by: ablinvadd 18930 ablsub2inv 18931 ablsubadd 18932 abladdsub 18935 ablpncan3 18937 ablsub32 18942 ablnnncan 18943 ablsubsub23 18945 eqgabl 18955 subgabl 18956 ablnsg 18967 lsmcomx 18976 qusabl 18985 frgpnabl 18995 ngplcan 23220 clmnegsubdi2 23709 clmvsubval2 23714 ncvspi 23760 r1pid 24753 abliso 30683 lindsunlem 31020 cnaddcom 36123 toycom 36124 lflsub 36218 lfladdcom 36223 |
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