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Theorem elmnc 41492
Description: Property of a monic polynomial. (Contributed by Stefan O'Rear, 5-Dec-2014.)
Assertion
Ref Expression
elmnc (𝑃 ∈ ( Monic β€˜π‘†) ↔ (𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1))

Proof of Theorem elmnc
Dummy variables 𝑠 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mnc 41489 . . . . 5 Monic = (𝑠 ∈ 𝒫 β„‚ ↦ {𝑝 ∈ (Polyβ€˜π‘ ) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1})
21dmmptss 6198 . . . 4 dom Monic βŠ† 𝒫 β„‚
3 elfvdm 6884 . . . 4 (𝑃 ∈ ( Monic β€˜π‘†) β†’ 𝑆 ∈ dom Monic )
42, 3sselid 3947 . . 3 (𝑃 ∈ ( Monic β€˜π‘†) β†’ 𝑆 ∈ 𝒫 β„‚)
54elpwid 4574 . 2 (𝑃 ∈ ( Monic β€˜π‘†) β†’ 𝑆 βŠ† β„‚)
6 plybss 25571 . . 3 (𝑃 ∈ (Polyβ€˜π‘†) β†’ 𝑆 βŠ† β„‚)
76adantr 482 . 2 ((𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1) β†’ 𝑆 βŠ† β„‚)
8 cnex 11139 . . . . . 6 β„‚ ∈ V
98elpw2 5307 . . . . 5 (𝑆 ∈ 𝒫 β„‚ ↔ 𝑆 βŠ† β„‚)
10 fveq2 6847 . . . . . . 7 (𝑠 = 𝑆 β†’ (Polyβ€˜π‘ ) = (Polyβ€˜π‘†))
11 rabeq 3424 . . . . . . 7 ((Polyβ€˜π‘ ) = (Polyβ€˜π‘†) β†’ {𝑝 ∈ (Polyβ€˜π‘ ) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1} = {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1})
1210, 11syl 17 . . . . . 6 (𝑠 = 𝑆 β†’ {𝑝 ∈ (Polyβ€˜π‘ ) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1} = {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1})
13 fvex 6860 . . . . . . 7 (Polyβ€˜π‘†) ∈ V
1413rabex 5294 . . . . . 6 {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1} ∈ V
1512, 1, 14fvmpt 6953 . . . . 5 (𝑆 ∈ 𝒫 β„‚ β†’ ( Monic β€˜π‘†) = {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1})
169, 15sylbir 234 . . . 4 (𝑆 βŠ† β„‚ β†’ ( Monic β€˜π‘†) = {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1})
1716eleq2d 2824 . . 3 (𝑆 βŠ† β„‚ β†’ (𝑃 ∈ ( Monic β€˜π‘†) ↔ 𝑃 ∈ {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1}))
18 fveq2 6847 . . . . . 6 (𝑝 = 𝑃 β†’ (coeffβ€˜π‘) = (coeffβ€˜π‘ƒ))
19 fveq2 6847 . . . . . 6 (𝑝 = 𝑃 β†’ (degβ€˜π‘) = (degβ€˜π‘ƒ))
2018, 19fveq12d 6854 . . . . 5 (𝑝 = 𝑃 β†’ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)))
2120eqeq1d 2739 . . . 4 (𝑝 = 𝑃 β†’ (((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1 ↔ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1))
2221elrab 3650 . . 3 (𝑃 ∈ {𝑝 ∈ (Polyβ€˜π‘†) ∣ ((coeffβ€˜π‘)β€˜(degβ€˜π‘)) = 1} ↔ (𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1))
2317, 22bitrdi 287 . 2 (𝑆 βŠ† β„‚ β†’ (𝑃 ∈ ( Monic β€˜π‘†) ↔ (𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1)))
245, 7, 23pm5.21nii 380 1 (𝑃 ∈ ( Monic β€˜π‘†) ↔ (𝑃 ∈ (Polyβ€˜π‘†) ∧ ((coeffβ€˜π‘ƒ)β€˜(degβ€˜π‘ƒ)) = 1))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {crab 3410   βŠ† wss 3915  π’« cpw 4565  dom cdm 5638  β€˜cfv 6501  β„‚cc 11056  1c1 11059  Polycply 25561  coeffccoe 25563  degcdgr 25564   Monic cmnc 41487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-cnex 11114
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fv 6509  df-ply 25565  df-mnc 41489
This theorem is referenced by:  mncply  41493  mnccoe  41494
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