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Mirrors > Home > MPE Home > Th. List > isablo | Structured version Visualization version GIF version |
Description: The predicate "is an Abelian (commutative) group operation." (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isabl.1 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
isablo | ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneq 5799 | . . . . 5 ⊢ (𝑔 = 𝐺 → ran 𝑔 = ran 𝐺) | |
2 | isabl.1 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
3 | 1, 2 | syl6eqr 2871 | . . . 4 ⊢ (𝑔 = 𝐺 → ran 𝑔 = 𝑋) |
4 | raleq 3403 | . . . . 5 ⊢ (ran 𝑔 = 𝑋 → (∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥))) | |
5 | 4 | raleqbi1dv 3401 | . . . 4 ⊢ (ran 𝑔 = 𝑋 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥))) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥))) |
7 | oveq 7151 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑥𝑔𝑦) = (𝑥𝐺𝑦)) | |
8 | oveq 7151 | . . . . 5 ⊢ (𝑔 = 𝐺 → (𝑦𝑔𝑥) = (𝑦𝐺𝑥)) | |
9 | 7, 8 | eqeq12d 2834 | . . . 4 ⊢ (𝑔 = 𝐺 → ((𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
10 | 9 | 2ralbidv 3196 | . . 3 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
11 | 6, 10 | bitrd 280 | . 2 ⊢ (𝑔 = 𝐺 → (∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
12 | df-ablo 28249 | . 2 ⊢ AbelOp = {𝑔 ∈ GrpOp ∣ ∀𝑥 ∈ ran 𝑔∀𝑦 ∈ ran 𝑔(𝑥𝑔𝑦) = (𝑦𝑔𝑥)} | |
13 | 11, 12 | elrab2 3680 | 1 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ran crn 5549 (class class class)co 7145 GrpOpcgr 28193 AbelOpcablo 28248 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-cnv 5556 df-dm 5558 df-rn 5559 df-iota 6307 df-fv 6356 df-ov 7148 df-ablo 28249 |
This theorem is referenced by: ablogrpo 28251 ablocom 28252 isabloi 28255 |
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