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Theorem elmnc 40453
 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.
StepHypRef Expression
1 df-mnc 40450 . . . . 5 Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
21dmmptss 6070 . . . 4 dom Monic ⊆ 𝒫 ℂ
3 elfvdm 6690 . . . 4 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
42, 3sseldi 3890 . . 3 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4505 . 2 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6 plybss 24890 . . 3 (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
76adantr 484 . 2 ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8 cnex 10656 . . . . . 6 ℂ ∈ V
98elpw2 5215 . . . . 5 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10 fveq2 6658 . . . . . . 7 (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11 rabeq 3396 . . . . . . 7 ((Poly‘𝑠) = (Poly‘𝑆) → {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
1210, 11syl 17 . . . . . 6 (𝑠 = 𝑆 → {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
13 fvex 6671 . . . . . . 7 (Poly‘𝑆) ∈ V
1413rabex 5202 . . . . . 6 {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
1512, 1, 14fvmpt 6759 . . . . 5 (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
169, 15sylbir 238 . . . 4 (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
1716eleq2d 2837 . . 3 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18 fveq2 6658 . . . . . 6 (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19 fveq2 6658 . . . . . 6 (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
2018, 19fveq12d 6665 . . . . 5 (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
2120eqeq1d 2760 . . . 4 (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2221elrab 3602 . . 3 (𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2317, 22bitrdi 290 . 2 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)))
245, 7, 23pm5.21nii 383 1 (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074   ⊆ wss 3858  𝒫 cpw 4494  dom cdm 5524  ‘cfv 6335  ℂcc 10573  1c1 10576  Polycply 24880  coeffccoe 24882  degcdgr 24883   Monic cmnc 40448 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-cnex 10631 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fv 6343  df-ply 24884  df-mnc 40450 This theorem is referenced by:  mncply  40454  mnccoe  40455
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