Theorem rrgval 20903

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 6892	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3		rrgval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑅)
4	2, 3	eqtr4di 2791	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5		fveq2 6892	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = (.r‘𝑅))
6		rrgval.t	. . . . . . . . . 10 ⊢ · = (.r‘𝑅)
7	5, 6	eqtr4di 2791	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (.r‘𝑟) = · )
8	7	oveqd 7426	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥(.r‘𝑟)𝑦) = (𝑥 · 𝑦))
9		fveq2 6892	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = (0g‘𝑅))
10		rrgval.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝑅)
11	9, 10	eqtr4di 2791	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘𝑟) = 0 )
12	8, 11	eqeq12d 2749	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) ↔ (𝑥 · 𝑦) = 0 ))
13	11	eqeq2d 2744	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝑦 = (0g‘𝑟) ↔ 𝑦 = 0 ))
14	12, 13	imbi12d 345	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟)) ↔ ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )))
15	4, 14	raleqbidv 3343	. . . . 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 3450	. . . 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 20899	. . . 4 ⊢ RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r‘𝑟)𝑦) = (0g‘𝑟) → 𝑦 = (0g‘𝑟))})
18	3	fvexi 6906	. . . . 5 ⊢ 𝐵 ∈ V
19	18	rabex 5333	. . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} ∈ V
20	16, 17, 19	fvmpt 6999	. . 3 ⊢ (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
21		fvprc 6884	. . . 4 ⊢ (¬ 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22		fvprc 6884	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑅) = ∅)
23	3, 22	eqtrid 2785	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
24	23	rabeqdv 3448	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )})
25		rab0 4383	. . . . 5 ⊢ {𝑥 ∈ ∅ ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = ∅
26	24, 25	eqtrdi 2789	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )} = ∅)
27	21, 26	eqtr4d 2776	. . 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 2761	1 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2107  ∀wral 3062  {crab 3433  Vcvv 3475  ∅c0 4323  ‘cfv 6544  (class class class)co 7409  Basecbs 17144  ._rcmulr 17198  0_gc0g 17385  RLRegcrlreg 20895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-rlreg 20899

This theorem is referenced by:  isrrg  20904  rrgeq0  20906  rrgss  20908