Theorem uc1pcl 24739

Description: Unitic polynomials are polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pcl.p	⊢ 𝑃 = (Poly1‘𝑅)
uc1pcl.b	⊢ 𝐵 = (Base‘𝑃)
uc1pcl.c	⊢ 𝐶 = (Unic1p‘𝑅)

Assertion

Ref	Expression
uc1pcl	⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵)

Proof of Theorem uc1pcl

Step	Hyp	Ref	Expression
1		uc1pcl.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
2		uc1pcl.b	. . 3 ⊢ 𝐵 = (Base‘𝑃)
3		eqid 2823	. . 3 ⊢ (0g‘𝑃) = (0g‘𝑃)
4		eqid 2823	. . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
5		uc1pcl.c	. . 3 ⊢ 𝐶 = (Unic1p‘𝑅)
6		eqid 2823	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
7	1, 2, 3, 4, 5, 6	isuc1p 24736	. 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ (0g‘𝑃) ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅)))
8	7	simp1bi 1141	1 ⊢ (𝐹 ∈ 𝐶 → 𝐹 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ≠ wne 3018  ‘cfv 6357  Basecbs 16485  0_gc0g 16715  Unitcui 19391  Poly₁cpl1 20347  coe₁cco1 20348   deg₁ cdg1 24650  Unic_1pcuc1p 24722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-slot 16489  df-base 16491  df-uc1p 24727

This theorem is referenced by:  uc1pdeg  24743  uc1pmon1p  24747  q1peqb  24750  r1pcl  24753  r1pdeglt  24754  r1pid  24755  dvdsq1p  24756  dvdsr1p  24757