Theorem elmnc 43239

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 43236	. . . . 5 ⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
2	1	dmmptss 6188	. . . 4 ⊢ dom Monic ⊆ 𝒫 ℂ
3		elfvdm 6856	. . . 4 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
4	2, 3	sselid 3927	. . 3 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
5	4	elpwid 4556	. 2 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6		plybss 26126	. . 3 ⊢ (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	6	adantr 480	. 2 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8		cnex 11087	. . . . . 6 ⊢ ℂ ∈ V
9	8	elpw2 5270	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10		fveq2 6822	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11		rabeq 3409	. . . . . . 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 6835	. . . . . . 7 ⊢ (Poly‘𝑆) ∈ V
14	13	rabex 5275	. . . . . 6 ⊢ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
15	12, 1, 14	fvmpt 6929	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
16	9, 15	sylbir 235	. . . 4 ⊢ (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
17	16	eleq2d 2817	. . 3 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18		fveq2 6822	. . . . . 6 ⊢ (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19		fveq2 6822	. . . . . 6 ⊢ (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
20	18, 19	fveq12d 6829	. . . . 5 ⊢ (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
21	20	eqeq1d 2733	. . . 4 ⊢ (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
22	21	elrab 3642	. . 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 378	1 ⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {crab 3395   ⊆ wss 3897  𝒫 cpw 4547  dom cdm 5614  ‘cfv 6481  ℂcc 11004  1c1 11007  Polycply 26116  coeffccoe 26118  degcdgr 26119   Monic cmnc 43234

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-cnex 11062

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-ply 26120  df-mnc 43236

This theorem is referenced by:  mncply  43240  mnccoe  43241