Theorem uc1pn0 24742

Description: Unitic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pn0.p	⊢ 𝑃 = (Poly1‘𝑅)
uc1pn0.z	⊢ 0 = (0g‘𝑃)
uc1pn0.c	⊢ 𝐶 = (Unic1p‘𝑅)

Assertion

Ref	Expression
uc1pn0	⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 )

Proof of Theorem uc1pn0

Step	Hyp	Ref	Expression
1		uc1pn0.p	. . 3 ⊢ 𝑃 = (Poly1‘𝑅)
2		eqid 2824	. . 3 ⊢ (Base‘𝑃) = (Base‘𝑃)
3		uc1pn0.z	. . 3 ⊢ 0 = (0g‘𝑃)
4		eqid 2824	. . 3 ⊢ ( deg1 ‘𝑅) = ( deg1 ‘𝑅)
5		uc1pn0.c	. . 3 ⊢ 𝐶 = (Unic1p‘𝑅)
6		eqid 2824	. . 3 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
7	1, 2, 3, 4, 5, 6	isuc1p 24737	. 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(( deg1 ‘𝑅)‘𝐹)) ∈ (Unit‘𝑅)))
8	7	simp2bi 1142	1 ⊢ (𝐹 ∈ 𝐶 → 𝐹 ≠ 0 )

Syntax hints:   → wi 4   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ‘cfv 6358  Basecbs 16486  0_gc0g 16716  Unitcui 19392  Poly₁cpl1 20348  coe₁cco1 20349   deg₁ cdg1 24651  Unic_1pcuc1p 24723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-iota 6317  df-fun 6360  df-fv 6366  df-slot 16490  df-base 16492  df-uc1p 24728

This theorem is referenced by:  uc1pdeg  24744  q1peqb  24751