Theorem isrrg 20472

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 7262	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
2	1	eqeq1d 2740	. . . 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 3120	. 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 20471	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}
10	4, 9	elrab2 3620	1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  ._rcmulr 16889  0_gc0g 17067  RLRegcrlreg 20463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-rlreg 20467

This theorem is referenced by:  rrgeq0i  20473  unitrrg  20477  isdomn2  20483