Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isabloi | Structured version Visualization version GIF version |
Description: Properties that determine an Abelian group operation. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isabli.1 | ⊢ 𝐺 ∈ GrpOp |
isabli.2 | ⊢ dom 𝐺 = (𝑋 × 𝑋) |
isabli.3 | ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) |
Ref | Expression |
---|---|
isabloi | ⊢ 𝐺 ∈ AbelOp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isabli.1 | . 2 ⊢ 𝐺 ∈ GrpOp | |
2 | isabli.3 | . . 3 ⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑥𝐺𝑦) = (𝑦𝐺𝑥)) | |
3 | 2 | rgen2 3203 | . 2 ⊢ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥) |
4 | isabli.2 | . . . 4 ⊢ dom 𝐺 = (𝑋 × 𝑋) | |
5 | 1, 4 | grporn 28298 | . . 3 ⊢ 𝑋 = ran 𝐺 |
6 | 5 | isablo 28323 | . 2 ⊢ (𝐺 ∈ AbelOp ↔ (𝐺 ∈ GrpOp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐺𝑦) = (𝑦𝐺𝑥))) |
7 | 1, 3, 6 | mpbir2an 709 | 1 ⊢ 𝐺 ∈ AbelOp |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 × cxp 5553 dom cdm 5555 (class class class)co 7156 GrpOpcgr 28266 AbelOpcablo 28321 |
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 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
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-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 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-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-ov 7159 df-grpo 28270 df-ablo 28322 |
This theorem is referenced by: cnaddabloOLD 28358 hilablo 28937 hhssabloi 29039 |
Copyright terms: Public domain | W3C validator |