Theorem iscom2 37319

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	⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ 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.

Step	Hyp	Ref	Expression
1		df-com2 37314	. . . 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 2811	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}))
4		rneq 5925	. . . 4 ⊢ (𝑥 = 𝐺 → ran 𝑥 = ran 𝐺)
5	4	raleqdv 3317	. . . 4 ⊢ (𝑥 = 𝐺 → (∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
6	4, 5	raleqbidv 3334	. . 3 ⊢ (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
7		oveq 7407	. . . . 5 ⊢ (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏))
8		oveq 7407	. . . . 5 ⊢ (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎))
9	7, 8	eqeq12d 2740	. . . 4 ⊢ (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
10	9	2ralbidv 3210	. . 3 ⊢ (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
11	6, 10	opelopabg 5528	. 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 279	1 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∀wral 3053  ⟨cop 4626  {copab 5200  ran crn 5667  (class class class)co 7401  Com2ccm2 37313

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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-cnv 5674  df-dm 5676  df-rn 5677  df-iota 6485  df-fv 6541  df-ov 7404  df-com2 37314

This theorem is referenced by:  iscrngo2  37321  iscringd  37322