Theorem isrrg 20663

Description: Membership in the set of left-regular elements. (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
isrrg	⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))

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

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

Proof of Theorem isrrg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7417	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
2	1	eqeq1d 2738	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
3	2	imbi1d 341	. . 3 ⊢ (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))
4	3	ralbidv 3164	. 2 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 ) ↔ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))
5		rrgval.e	. . 3 ⊢ 𝐸 = (RLReg‘𝑅)
6		rrgval.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
7		rrgval.t	. . 3 ⊢ · = (.r‘𝑅)
8		rrgval.z	. . 3 ⊢ 0 = (0g‘𝑅)
9	5, 6, 7, 8	rrgval 20662	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}
10	4, 9	elrab2 3679	1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3052  ‘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:  rrgeq0i  20664  unitrrg  20668  isdomn2  20676  isdomn2OLD  20677  rrgsubm  33283  zringidom  33571