Theorem elmnc 39726

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 39723	. . . . 5 ⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
2	1	dmmptss 6088	. . . 4 ⊢ dom Monic ⊆ 𝒫 ℂ
3		elfvdm 6695	. . . 4 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
4	2, 3	sseldi 3963	. . 3 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
5	4	elpwid 4551	. 2 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6		plybss 24776	. . 3 ⊢ (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	6	adantr 483	. 2 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8		cnex 10610	. . . . . 6 ⊢ ℂ ∈ V
9	8	elpw2 5239	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10		fveq2 6663	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11		rabeq 3482	. . . . . . 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 6676	. . . . . . 7 ⊢ (Poly‘𝑆) ∈ V
14	13	rabex 5226	. . . . . 6 ⊢ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
15	12, 1, 14	fvmpt 6761	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
16	9, 15	sylbir 237	. . . 4 ⊢ (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
17	16	eleq2d 2896	. . 3 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18		fveq2 6663	. . . . . 6 ⊢ (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19		fveq2 6663	. . . . . 6 ⊢ (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
20	18, 19	fveq12d 6670	. . . . 5 ⊢ (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
21	20	eqeq1d 2821	. . . 4 ⊢ (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
22	21	elrab 3678	. . 3 ⊢ (𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
23	17, 22	syl6bb 289	. 2 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)))
24	5, 7, 23	pm5.21nii 382	1 ⊢ (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  {crab 3140   ⊆ wss 3934  𝒫 cpw 4537  dom cdm 5548  ‘cfv 6348  ℂcc 10527  1c1 10530  Polycply 24766  coeffccoe 24768  degcdgr 24769   Monic cmnc 39721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-cnex 10585

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ply 24770  df-mnc 39723

This theorem is referenced by:  mncply  39727  mnccoe  39728