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Theorem elmnc 43125
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 43122 . . . . 5 Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
21dmmptss 6263 . . . 4 dom Monic ⊆ 𝒫 ℂ
3 elfvdm 6944 . . . 4 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
42, 3sselid 3993 . . 3 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4614 . 2 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6 plybss 26248 . . 3 (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
76adantr 480 . 2 ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8 cnex 11234 . . . . . 6 ℂ ∈ V
98elpw2 5340 . . . . 5 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10 fveq2 6907 . . . . . . 7 (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11 rabeq 3448 . . . . . . 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 6920 . . . . . . 7 (Poly‘𝑆) ∈ V
1413rabex 5345 . . . . . 6 {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
1512, 1, 14fvmpt 7016 . . . . 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 2825 . . 3 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18 fveq2 6907 . . . . . 6 (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19 fveq2 6907 . . . . . 6 (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
2018, 19fveq12d 6914 . . . . 5 (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
2120eqeq1d 2737 . . . 4 (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2221elrab 3695 . . 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 1537  wcel 2106  {crab 3433  wss 3963  𝒫 cpw 4605  dom cdm 5689  cfv 6563  cc 11151  1c1 11154  Polycply 26238  coeffccoe 26240  degcdgr 26241   Monic cmnc 43120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-cnex 11209
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ply 26242  df-mnc 43122
This theorem is referenced by:  mncply  43126  mnccoe  43127
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