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Theorem uc1pldg 24747
 Description: Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pldg.d 𝐷 = ( deg1𝑅)
uc1pldg.u 𝑈 = (Unit‘𝑅)
uc1pldg.c 𝐶 = (Unic1p𝑅)
Assertion
Ref Expression
uc1pldg (𝐹𝐶 → ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)

Proof of Theorem uc1pldg
StepHypRef Expression
1 eqid 2822 . . 3 (Poly1𝑅) = (Poly1𝑅)
2 eqid 2822 . . 3 (Base‘(Poly1𝑅)) = (Base‘(Poly1𝑅))
3 eqid 2822 . . 3 (0g‘(Poly1𝑅)) = (0g‘(Poly1𝑅))
4 uc1pldg.d . . 3 𝐷 = ( deg1𝑅)
5 uc1pldg.c . . 3 𝐶 = (Unic1p𝑅)
6 uc1pldg.u . . 3 𝑈 = (Unit‘𝑅)
71, 2, 3, 4, 5, 6isuc1p 24739 . 2 (𝐹𝐶 ↔ (𝐹 ∈ (Base‘(Poly1𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1𝑅)) ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
87simp3bi 1144 1 (𝐹𝐶 → ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2114   ≠ wne 3011  ‘cfv 6334  Basecbs 16474  0gc0g 16704  Unitcui 19383  Poly1cpl1 20804  coe1cco1 20805   deg1 cdg1 24653  Unic1pcuc1p 24725 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-slot 16478  df-base 16480  df-uc1p 24730 This theorem is referenced by:  uc1pmon1p  24750  q1peqb  24753  fta1glem1  24764  ig1peu  24770
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