Theorem elmnc 36725

Description: Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)

Assertion

Ref	Expression
elmnc	⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))

Proof of Theorem elmnc

Dummy variables 𝑠 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mnc 36722	. . . . 5 ⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
2	1	dmmptss 5548	. . . 4 ⊢ dom Monic ⊆ 𝒫 ℂ
3		elfvdm 6130	. . . 4 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
4	2, 3	sseldi 3566	. . 3 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
5	4	elpwid 4118	. 2 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6		plybss 23754	. . 3 ⊢ (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	6	adantr 480	. 2 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8		cnex 9896	. . . . . 6 ⊢ ℂ ∈ V
9	8	elpw2 4755	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10		fveq2 6103	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11		rabeq 3166	. . . . . . 7 ⊢ ((Poly‘𝑠) = (Poly‘𝑆) → {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
12	10, 11	syl 17	. . . . . 6 ⊢ (𝑠 = 𝑆 → {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
13		fvex 6113	. . . . . . 7 ⊢ (Poly‘𝑆) ∈ V
14	13	rabex 4740	. . . . . 6 ⊢ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
15	12, 1, 14	fvmpt 6191	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
16	9, 15	sylbir 224	. . . 4 ⊢ (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
17	16	eleq2d 2673	. . 3 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18		fveq2 6103	. . . . . 6 ⊢ (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19		fveq2 6103	. . . . . 6 ⊢ (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
20	18, 19	fveq12d 6109	. . . . 5 ⊢ (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
21	20	eqeq1d 2612	. . . 4 ⊢ (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
22	21	elrab 3331	. . 3 ⊢ (𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
23	17, 22	syl6bb 275	. 2 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)))
24	5, 7, 23	pm5.21nii 367	1 ⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  dom cdm 5038  ‘cfv 5804  ℂcc 9813  1c1 9816  Polycply 23744  coeffccoe 23746  degcdgr 23747   Monic cmnc 36720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-cnex 9871

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ply 23748  df-mnc 36722

This theorem is referenced by:  mncply  36726  mnccoe  36727