Theorem uc1pldg 24128

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 2771	. . 3 ⊢ (Poly1‘𝑅) = (Poly1‘𝑅)
2		eqid 2771	. . 3 ⊢ (Base‘(Poly1‘𝑅)) = (Base‘(Poly1‘𝑅))
3		eqid 2771	. . 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 24120	. 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘(Poly1‘𝑅)) ∧ 𝐹 ≠ (0g‘(Poly1‘𝑅)) ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))
8	7	simp3bi 1141	1 ⊢ (𝐹 ∈ 𝐶 → ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1631   ∈ wcel 2145   ≠ wne 2943  ‘cfv 6031  Basecbs 16064  0_gc0g 16308  Unitcui 18847  Poly₁cpl1 19762  coe₁cco1 19763   deg₁ cdg1 24034  Unic_1pcuc1p 24106

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 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034

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 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-slot 16068  df-base 16070  df-uc1p 24111

This theorem is referenced by:  uc1pmon1p  24131  q1peqb  24134  fta1glem1  24145  ig1peu  24151