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Theorem iscom2 35143
Description: A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (New usage is discouraged.)
Assertion
Ref Expression
iscom2 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
Distinct variable groups:   𝐺,𝑎,𝑏   𝐻,𝑎,𝑏
Allowed substitution hints:   𝐴(𝑎,𝑏)   𝐵(𝑎,𝑏)

Proof of Theorem iscom2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-com2 35138 . . . 4 Com2 = {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}
21a1i 11 . . 3 ((𝐺𝐴𝐻𝐵) → Com2 = {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)})
32eleq2d 2902 . 2 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}))
4 rneq 5804 . . . 4 (𝑥 = 𝐺 → ran 𝑥 = ran 𝐺)
54raleqdv 3420 . . . 4 (𝑥 = 𝐺 → (∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
64, 5raleqbidv 3406 . . 3 (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
7 oveq 7157 . . . . 5 (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏))
8 oveq 7157 . . . . 5 (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎))
97, 8eqeq12d 2841 . . . 4 (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
1092ralbidv 3203 . . 3 (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
116, 10opelopabg 5421 . 2 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)} ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
123, 11bitrd 280 1 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  wral 3142  cop 4569  {copab 5124  ran crn 5554  (class class class)co 7151  Com2ccm2 35137
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-cnv 5561  df-dm 5563  df-rn 5564  df-iota 6311  df-fv 6359  df-ov 7154  df-com2 35138
This theorem is referenced by:  iscrngo2  35145  iscringd  35146
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