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Mirrors > Home > MPE Home > Th. List > ablocom | Structured version Visualization version GIF version |
Description: An Abelian group operation is commutative. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ablcom.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
ablocom | ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ablcom.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
2 | 1 | isablo 28326 | . . . 4 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
3 | 2 | simprbi 499 | . . 3 ⊢ (𝐺 ∈ AbelOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) |
4 | oveq1 7166 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦)) | |
5 | oveq2 7167 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴)) | |
6 | 4, 5 | eqeq12d 2840 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴))) |
7 | oveq2 7167 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) | |
8 | oveq1 7166 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴)) | |
9 | 7, 8 | eqeq12d 2840 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴))) |
10 | 6, 9 | rspc2v 3636 | . . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))) |
11 | 3, 10 | syl5com 31 | . 2 ⊢ (𝐺 ∈ AbelOp → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴))) |
12 | 11 | 3impib 1112 | 1 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ran crn 5559 (class class class)co 7159 GrpOpcgr 28269 AbelOpcablo 28324 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-cnv 5566 df-dm 5568 df-rn 5569 df-iota 6317 df-fv 6366 df-ov 7162 df-ablo 28325 |
This theorem is referenced by: ablo32 28329 ablomuldiv 28332 ablodiv32 28335 nvcom 28401 rngocom 35195 iscringd 35280 |
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