Theorem iscom2 38458

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 38453	. . . 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 2847	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}))
4		rneq 5910	. . . 4 ⊢ (𝑥 = 𝐺 → ran 𝑥 = ran 𝐺)
5	4	raleqdv 3319	. . . 4 ⊢ (𝑥 = 𝐺 → (∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
6	4, 5	raleqbidv 3335	. . 3 ⊢ (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
7		oveq 7398	. . . . 5 ⊢ (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏))
8		oveq 7398	. . . . 5 ⊢ (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎))
9	7, 8	eqeq12d 2777	. . . 4 ⊢ (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
10	9	2ralbidv 3225	. . 3 ⊢ (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
11	6, 10	opelopabg 5507	. 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 281	1 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ⟨cop 4587  {copab 5161  ran crn 5646  (class class class)co 7392  Com2ccm2 38452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-cnv 5653  df-dm 5655  df-rn 5656  df-iota 6473  df-fv 6525  df-ov 7395  df-com2 38453

This theorem is referenced by:  iscrngo2  38460  iscringd  38461