Theorem elmnc 39785

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 39782	. . . . 5 ⊢ Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
2	1	dmmptss 6095	. . . 4 ⊢ dom Monic ⊆ 𝒫 ℂ
3		elfvdm 6702	. . . 4 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
4	2, 3	sseldi 3965	. . 3 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
5	4	elpwid 4550	. 2 ⊢ (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6		plybss 24784	. . 3 ⊢ (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
7	6	adantr 483	. 2 ⊢ ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8		cnex 10618	. . . . . 6 ⊢ ℂ ∈ V
9	8	elpw2 5248	. . . . 5 ⊢ (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10		fveq2 6670	. . . . . . 7 ⊢ (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11		rabeq 3483	. . . . . . 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 6683	. . . . . . 7 ⊢ (Poly‘𝑆) ∈ V
14	13	rabex 5235	. . . . . 6 ⊢ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
15	12, 1, 14	fvmpt 6768	. . . . 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 2898	. . 3 ⊢ (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18		fveq2 6670	. . . . . 6 ⊢ (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19		fveq2 6670	. . . . . 6 ⊢ (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
20	18, 19	fveq12d 6677	. . . . 5 ⊢ (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
21	20	eqeq1d 2823	. . . 4 ⊢ (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
22	21	elrab 3680	. . 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 1537   ∈ wcel 2114  {crab 3142   ⊆ wss 3936  𝒫 cpw 4539  dom cdm 5555  ‘cfv 6355  ℂcc 10535  1c1 10538  Polycply 24774  coeffccoe 24776  degcdgr 24777   Monic cmnc 39780

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-cnex 10593

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fv 6363  df-ply 24778  df-mnc 39782

This theorem is referenced by:  mncply  39786  mnccoe  39787