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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 18199 | . 2 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd) | |
2 | ablcom.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | ablcom.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | cmncom 18209 | . 2 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
5 | 1, 4 | syl3an1 1359 | 1 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 CMndccmn 18193 Abelcabl 18194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-cmn 18195 df-abl 18196 |
This theorem is referenced by: ablinvadd 18215 ablsub2inv 18216 ablsubadd 18217 abladdsub 18220 ablpncan3 18222 ablsub32 18227 ablnnncan 18228 ablsubsub23 18230 eqgabl 18240 subgabl 18241 ablnsg 18250 lsmcomx 18259 qusabl 18268 frgpnabl 18278 ngplcan 22415 clmnegsubdi2 22905 clmvsubval2 22910 ncvspi 22956 r1pid 23919 abliso 29696 cnaddcom 34259 toycom 34260 lflsub 34354 lfladdcom 34359 |
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