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Mathbox for Jeff Madsen |
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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 35428 | . . . 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 2875 | . 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (〈𝐺, 𝐻〉 ∈ Com2 ↔ 〈𝐺, 𝐻〉 ∈ {〈𝑥, 𝑦〉 ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)})) |
4 | rneq 5770 | . . . 4 ⊢ (𝑥 = 𝐺 → ran 𝑥 = ran 𝐺) | |
5 | 4 | raleqdv 3364 | . . . 4 ⊢ (𝑥 = 𝐺 → (∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎))) |
6 | 4, 5 | raleqbidv 3354 | . . 3 ⊢ (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎))) |
7 | oveq 7141 | . . . . 5 ⊢ (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏)) | |
8 | oveq 7141 | . . . . 5 ⊢ (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎)) | |
9 | 7, 8 | eqeq12d 2814 | . . . 4 ⊢ (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
10 | 9 | 2ralbidv 3164 | . . 3 ⊢ (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎))) |
11 | 6, 10 | opelopabg 5390 | . 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 282 | 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 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 〈cop 4531 {copab 5092 ran crn 5520 (class class class)co 7135 Com2ccm2 35427 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-iota 6283 df-fv 6332 df-ov 7138 df-com2 35428 |
This theorem is referenced by: iscrngo2 35435 iscringd 35436 |
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