Theorem rrgval 20662

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 6881	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		rrgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2789	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
6		rrgval.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
7	5, 6	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
8	7	oveqd 7427	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
9		fveq2 6881	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
10		rrgval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
11	9, 10	eqtr4di 2789	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
12	8, 11	eqeq12d 2752	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 ))
13	11	eqeq2d 2747	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦 = (0g‘𝑟) ↔ 𝑦 = 0 ))
14	12, 13	imbi12d 344	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )))
15	4, 14	raleqbidv 3329	. . . . 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 3439	. . . 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 20659	. . . 4 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
18	3	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
19	18	rabex 5314	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} ∈ V
20	16, 17, 19	fvmpt 6991	. . 3 ⊢ (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
21		fvprc 6873	. . . 4 ⊢ (¬ 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22		fvprc 6873	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
23	3, 22	eqtrid 2783	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
24	23	rabeqdv 3436	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
25		rab0 4366	. . . . 5 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = ∅
26	24, 25	eqtrdi 2787	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = ∅)
27	21, 26	eqtr4d 2774	. . 3 ⊢ (¬ 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
28	20, 27	pm2.61i 182	. 2 ⊢ (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}
29	1, 28	eqtri 2759	1 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3052  {crab 3420  Vcvv 3464  ∅c0 4313  ‘cfv 6536  (class class class)co 7410  Basecbs 17233  ._rcmulr 17277  0_gc0g 17458  RLRegcrlreg 20656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6489  df-fun 6538  df-fv 6544  df-ov 7413  df-rlreg 20659

This theorem is referenced by:  isrrg  20663  rrgeq0  20665  rrgss  20667