Theorem uc1pldg 24314

Description: Unitic polynomials have unit leading coefficients. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pldg.d	⊢ 𝐷 = ( deg1 ‘𝑅)
uc1pldg.u	⊢ 𝑈 = (Unit‘𝑅)
uc1pldg.c	⊢ 𝐶 = (Unic1p‘𝑅)

Assertion

Ref	Expression
uc1pldg	⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)

Proof of Theorem uc1pldg

Step	Hyp	Ref	Expression
1		eqid 2825	. . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
2		eqid 2825	. . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
3		eqid 2825	. . 3 ⊢ (0g‘(Poly1‘𝑅)) = (0g‘(Poly1‘𝑅))
4		uc1pldg.d	. . 3 ⊢ 𝐷 = ( deg1 ‘𝑅)
5		uc1pldg.c	. . 3 ⊢ 𝐶 = (Unic1p‘𝑅)
6		uc1pldg.u	. . 3 ⊢ 𝑈 = (Unit‘𝑅)
7	1, 2, 3, 4, 5, 6	isuc1p 24306	. 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))
8	7	simp3bi 1181	1 ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1656   ∈ wcel 2164   ≠ wne 2999  ‘cfv 6127  Basecbs 16229  0_gc0g 16460  Unitcui 19000  Poly₁cpl1 19914  coe₁cco1 19915   deg₁ cdg1 24220  Unic_1pcuc1p 24292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-slot 16233  df-base 16235  df-uc1p 24297

This theorem is referenced by:  uc1pmon1p  24317  q1peqb  24320  fta1glem1  24331  ig1peu  24337