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Theorem elmnc 43153
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 43150 . . . . 5 Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
21dmmptss 6260 . . . 4 dom Monic ⊆ 𝒫 ℂ
3 elfvdm 6942 . . . 4 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
42, 3sselid 3980 . . 3 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4608 . 2 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6 plybss 26234 . . 3 (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
76adantr 480 . 2 ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8 cnex 11237 . . . . . 6 ℂ ∈ V
98elpw2 5333 . . . . 5 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10 fveq2 6905 . . . . . . 7 (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11 rabeq 3450 . . . . . . 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 6918 . . . . . . 7 (Poly‘𝑆) ∈ V
1413rabex 5338 . . . . . 6 {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
1512, 1, 14fvmpt 7015 . . . . 5 (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
169, 15sylbir 235 . . . 4 (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
1716eleq2d 2826 . . 3 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18 fveq2 6905 . . . . . 6 (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19 fveq2 6905 . . . . . 6 (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
2018, 19fveq12d 6912 . . . . 5 (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
2120eqeq1d 2738 . . . 4 (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2221elrab 3691 . . 3 (𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2317, 22bitrdi 287 . 2 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)))
245, 7, 23pm5.21nii 378 1 (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1539  wcel 2107  {crab 3435  wss 3950  𝒫 cpw 4599  dom cdm 5684  cfv 6560  cc 11154  1c1 11157  Polycply 26224  coeffccoe 26226  degcdgr 26227   Monic cmnc 43148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-cnex 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ply 26228  df-mnc 43150
This theorem is referenced by:  mncply  43154  mnccoe  43155
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