Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > iscom2 | Structured version Visualization version GIF version |
Description: A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.) |
Ref | Expression |
---|---|
iscom2 | ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (〈𝐺, 𝐻〉 ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-com2 32959 | . . . 4 ⊢ Com2 = {〈𝑥, 𝑦〉 ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)} | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → Com2 = {〈𝑥, 𝑦〉 ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}) |
3 | 2 | eleq2d 2673 | . 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (〈𝐺, 𝐻〉 ∈ Com2 ↔ 〈𝐺, 𝐻〉 ∈ {〈𝑥, 𝑦〉 ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)})) |
4 | rneq 5272 | . . . 4 ⊢ (𝑥 = 𝐺 → ran 𝑥 = ran 𝐺) | |
5 | 4 | raleqdv 3121 | . . . 4 ⊢ (𝑥 = 𝐺 → (∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎))) |
6 | 4, 5 | raleqbidv 3129 | . . 3 ⊢ (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎))) |
7 | oveq 6555 | . . . . 5 ⊢ (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏)) | |
8 | oveq 6555 | . . . . 5 ⊢ (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎)) | |
9 | 7, 8 | eqeq12d 2625 | . . . 4 ⊢ (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
10 | 9 | 2ralbidv 2972 | . . 3 ⊢ (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
11 | 6, 10 | opelopabg 4918 | . 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (〈𝐺, 𝐻〉 ∈ {〈𝑥, 𝑦〉 ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)} ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
12 | 3, 11 | bitrd 267 | 1 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (〈𝐺, 𝐻〉 ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 〈cop 4131 {copab 4642 ran crn 5039 (class class class)co 6549 Com2ccm2 32958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-cnv 5046 df-dm 5048 df-rn 5049 df-iota 5768 df-fv 5812 df-ov 6552 df-com2 32959 |
This theorem is referenced by: iscrngo2 32966 iscringd 32967 |
Copyright terms: Public domain | W3C validator |