Theorem rrgval 20778

Description: Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)

Hypotheses

Ref	Expression
rrgval.e	⊢ 𝐸 = (RLReg‘𝑅)
rrgval.b	⊢ 𝐵 = (Base‘𝑅)
rrgval.t	⊢ · = (.r‘𝑅)
rrgval.z	⊢ 0 = (0g‘𝑅)

Assertion

Ref	Expression
rrgval	⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦

Allowed substitution hints:   · (𝑥,𝑦)   𝐸(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem rrgval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rrgval.e	. 2 ⊢ 𝐸 = (RLReg‘𝑅)
2		fveq2 6879	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		rrgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2822	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
6		rrgval.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
7	5, 6	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
8	7	oveqd 7425	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
9		fveq2 6879	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
10		rrgval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
11	9, 10	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
12	8, 11	eqeq12d 2785	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 ))
13	11	eqeq2d 2780	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦 = (0g‘𝑟) ↔ 𝑦 = 0 ))
14	12, 13	imbi12d 347	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )))
15	4, 14	raleqbidv 3345	. . . . 5 ⊢ (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )))
16	4, 15	rabeqbidv 3441	. . . 4 ⊢ (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))} = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
17		df-rlreg 20775	. . . 4 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
18	3	fvexi 6893	. . . . 5 ⊢ 𝐵 ∈ V
19	18	rabex 5307	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} ∈ V
20	16, 17, 19	fvmpt 6987	. . 3 ⊢ (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
21		fvprc 6871	. . . 4 ⊢ (¬ 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22		fvprc 6871	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
23	3, 22	eqtrid 2816	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
24	23	rabeqdv 3438	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
25		rab0 4348	. . . . 5 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = ∅
26	24, 25	eqtrdi 2820	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = ∅)
27	21, 26	eqtr4d 2807	. . 3 ⊢ (¬ 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
28	20, 27	pm2.61i 184	. 2 ⊢ (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}
29	1, 28	eqtri 2792	1 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085  {crab 3423  Vcvv 3463  ∅c0 4294  ‘cfv 6534  (class class class)co 7408  Basecbs 17265  ._rcmulr 17307  0_gc0g 17488  RLRegcrlreg 20772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6490  df-fun 6536  df-fv 6542  df-ov 7411  df-rlreg 20775

This theorem is referenced by:  isrrg  20779  rrgeq0  20781  rrgss  20783