Theorem isrrg 20622

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 7362	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
2	1	eqeq1d 2735	. . . 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 3156	. 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 20621	. 2 ⊢ 𝐸 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 · 𝑦) = 0 → 𝑦 = 0 )}
10	4, 9	elrab2 3646	1 ⊢ (𝑋 ∈ 𝐸 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑋 · 𝑦) = 0 → 𝑦 = 0 )))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  ._rcmulr 17169  0_gc0g 17350  RLRegcrlreg 20615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-rlreg 20618

This theorem is referenced by:  rrgeq0i  20623  unitrrg  20627  isdomn2  20635  isdomn2OLD  20636  rrgsubm  33294  zringidom  33560