Theorem elmnc 40961

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 40958	. . . . 5 ⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
2	1	dmmptss 6144	. . . 4 ⊢ dom Monic ⊆ 𝒫 ℂ
3		elfvdm 6806	. . . 4 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
4	2, 3	sselid 3919	. . 3 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
5	4	elpwid 4544	. 2 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6		plybss 25355	. . 3 ⊢ (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	6	adantr 481	. 2 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8		cnex 10952	. . . . . 6 ⊢ ℂ ∈ V
9	8	elpw2 5269	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10		fveq2 6774	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11		rabeq 3418	. . . . . . 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 6787	. . . . . . 7 ⊢ (Poly‘𝑆) ∈ V
14	13	rabex 5256	. . . . . 6 ⊢ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
15	12, 1, 14	fvmpt 6875	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
16	9, 15	sylbir 234	. . . 4 ⊢ (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
17	16	eleq2d 2824	. . 3 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18		fveq2 6774	. . . . . 6 ⊢ (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19		fveq2 6774	. . . . . 6 ⊢ (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
20	18, 19	fveq12d 6781	. . . . 5 ⊢ (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
21	20	eqeq1d 2740	. . . 4 ⊢ (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
22	21	elrab 3624	. . 3 ⊢ (𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
23	17, 22	bitrdi 287	. 2 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)))
24	5, 7, 23	pm5.21nii 380	1 ⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  {crab 3068   ⊆ wss 3887  𝒫 cpw 4533  dom cdm 5589  ‘cfv 6433  ℂcc 10869  1c1 10872  Polycply 25345  coeffccoe 25347  degcdgr 25348   Monic cmnc 40956

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-cnex 10927

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fv 6441  df-ply 25349  df-mnc 40958

This theorem is referenced by:  mncply  40962  mnccoe  40963