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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 27400 | . . . 4 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
3 | 2 | simprbi 480 | . . 3 ⊢ (𝐺 ∈ AbelOp → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) |
4 | oveq1 6657 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥𝐺𝑦) = (𝐴𝐺𝑦)) | |
5 | oveq2 6658 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑦𝐺𝑥) = (𝑦𝐺𝐴)) | |
6 | 4, 5 | eqeq12d 2637 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥𝐺𝑦) = (𝑦𝐺𝑥) ↔ (𝐴𝐺𝑦) = (𝑦𝐺𝐴))) |
7 | oveq2 6658 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴𝐺𝑦) = (𝐴𝐺𝐵)) | |
8 | oveq1 6657 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦𝐺𝐴) = (𝐵𝐺𝐴)) | |
9 | 7, 8 | eqeq12d 2637 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝐴𝐺𝑦) = (𝑦𝐺𝐴) ↔ (𝐴𝐺𝐵) = (𝐵𝐺𝐴))) |
10 | 6, 9 | rspc2v 3322 | . . 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 1262 | 1 ⊢ ((𝐺 ∈ AbelOp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐺𝐵) = (𝐵𝐺𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ran crn 5115 (class class class)co 6650 GrpOpcgr 27343 AbelOpcablo 27398 |
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-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 df-iota 5851 df-fv 5896 df-ov 6653 df-ablo 27399 |
This theorem is referenced by: ablo32 27403 ablomuldiv 27406 ablodiv32 27409 nvcom 27476 rngocom 33712 iscringd 33797 |
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