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Theorem iscom2 34103
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 34098 . . . 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 2871 . 2 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}))
4 rneq 5552 . . . 4 (𝑥 = 𝐺 → ran 𝑥 = ran 𝐺)
54raleqdv 3333 . . . 4 (𝑥 = 𝐺 → (∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
64, 5raleqbidv 3341 . . 3 (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
7 oveq 6876 . . . . 5 (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏))
8 oveq 6876 . . . . 5 (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎))
97, 8eqeq12d 2821 . . . 4 (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
1092ralbidv 3177 . . 3 (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
116, 10opelopabg 5188 . 2 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)} ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
123, 11bitrd 270 1 ((𝐺𝐴𝐻𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  cop 4376  {copab 4906  ran crn 5312  (class class class)co 6870  Com2ccm2 34097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-cnv 5319  df-dm 5321  df-rn 5322  df-iota 6060  df-fv 6105  df-ov 6873  df-com2 34098
This theorem is referenced by:  iscrngo2  34105  iscringd  34106
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