Theorem iscom2 39022

Description: Obsolete theorem, used (as lemma) in other obsolete theorems only. A device to add commutativity to various sorts of rings. (Contributed by FL, 6-Sep-2009.) (Proof modification is discouraged.) (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 39017	. . . 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 2846	. 2 ⊢ ((𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵) → (⟨𝐺, 𝐻⟩ ∈ Com2 ↔ ⟨𝐺, 𝐻⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎)}))
4		rneq 5917	. . . 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 3334	. . 3 ⊢ (𝑥 = 𝐺 → (∀𝑎 ∈ ran 𝑥∀𝑏 ∈ ran 𝑥(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎)))
7		oveq 7426	. . . . 5 ⊢ (𝑦 = 𝐻 → (𝑎𝑦𝑏) = (𝑎𝐻𝑏))
8		oveq 7426	. . . . 5 ⊢ (𝑦 = 𝐻 → (𝑏𝑦𝑎) = (𝑏𝐻𝑎))
9	7, 8	eqeq12d 2776	. . . 4 ⊢ (𝑦 = 𝐻 → ((𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ (𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
10	9	2ralbidv 3226	. . 3 ⊢ (𝑦 = 𝐻 → (∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝑦𝑏) = (𝑏𝑦𝑎) ↔ ∀𝑎 ∈ ran 𝐺∀𝑏 ∈ ran 𝐺(𝑎𝐻𝑏) = (𝑏𝐻𝑎)))
11	6, 10	opelopabg 5512	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ⟨cop 4834  {copab 5166  ran crn 5651  (class class class)co 7420  Com2ccm2 39016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-cnv 5658  df-dm 5660  df-rn 5661  df-iota 6494  df-fv 6546  df-ov 7423  df-com2 39017

This theorem is used by:  iscrngo2  39024  iscringd  39025