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Theorem elmnc 38232
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 38229 . . . . 5 Monic = (𝑠 ∈ 𝒫 ℂ ↦ {𝑝 ∈ (Poly‘𝑠) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
21dmmptss 5774 . . . 4 dom Monic ⊆ 𝒫 ℂ
3 elfvdm 6363 . . . 4 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ dom Monic )
42, 3sseldi 3750 . . 3 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ∈ 𝒫 ℂ)
54elpwid 4310 . 2 (𝑃 ∈ ( Monic ‘𝑆) → 𝑆 ⊆ ℂ)
6 plybss 24170 . . 3 (𝑃 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ)
76adantr 466 . 2 ((𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1) → 𝑆 ⊆ ℂ)
8 cnex 10223 . . . . . 6 ℂ ∈ V
98elpw2 4960 . . . . 5 (𝑆 ∈ 𝒫 ℂ ↔ 𝑆 ⊆ ℂ)
10 fveq2 6333 . . . . . . 7 (𝑠 = 𝑆 → (Poly‘𝑠) = (Poly‘𝑆))
11 rabeq 3342 . . . . . . 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 6344 . . . . . . 7 (Poly‘𝑆) ∈ V
1413rabex 4947 . . . . . 6 {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ∈ V
1512, 1, 14fvmpt 6426 . . . . 5 (𝑆 ∈ 𝒫 ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
169, 15sylbir 225 . . . 4 (𝑆 ⊆ ℂ → ( Monic ‘𝑆) = {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1})
1716eleq2d 2836 . . 3 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ 𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1}))
18 fveq2 6333 . . . . . 6 (𝑝 = 𝑃 → (coeff‘𝑝) = (coeff‘𝑃))
19 fveq2 6333 . . . . . 6 (𝑝 = 𝑃 → (deg‘𝑝) = (deg‘𝑃))
2018, 19fveq12d 6340 . . . . 5 (𝑝 = 𝑃 → ((coeff‘𝑝)‘(deg‘𝑝)) = ((coeff‘𝑃)‘(deg‘𝑃)))
2120eqeq1d 2773 . . . 4 (𝑝 = 𝑃 → (((coeff‘𝑝)‘(deg‘𝑝)) = 1 ↔ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2221elrab 3515 . . 3 (𝑃 ∈ {𝑝 ∈ (Poly‘𝑆) ∣ ((coeff‘𝑝)‘(deg‘𝑝)) = 1} ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
2317, 22syl6bb 276 . 2 (𝑆 ⊆ ℂ → (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1)))
245, 7, 23pm5.21nii 367 1 (𝑃 ∈ ( Monic ‘𝑆) ↔ (𝑃 ∈ (Poly‘𝑆) ∧ ((coeff‘𝑃)‘(deg‘𝑃)) = 1))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  {crab 3065  wss 3723  𝒫 cpw 4298  dom cdm 5250  cfv 6030  cc 10140  1c1 10143  Polycply 24160  coeffccoe 24162  degcdgr 24163   Monic cmnc 38227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-cnex 10198
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fv 6038  df-ply 24164  df-mnc 38229
This theorem is referenced by:  mncply  38233  mnccoe  38234
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