Theorem isuc1p 24885

Description: Being a unitic polynomial. (Contributed by Stefan O'Rear, 28-Mar-2015.)

Hypotheses

Ref	Expression
uc1pval.p	⊢ 𝑃 = (Poly1‘𝑅)
uc1pval.b	⊢ 𝐵 = (Base‘𝑃)
uc1pval.z	⊢ 0 = (0g‘𝑃)
uc1pval.d	⊢ 𝐷 = ( deg1 ‘𝑅)
uc1pval.c	⊢ 𝐶 = (Unic1p‘𝑅)
uc1pval.u	⊢ 𝑈 = (Unit‘𝑅)

Assertion

Ref	Expression
isuc1p	⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))

Proof of Theorem isuc1p

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neeq1 2996	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓 ≠ 0 ↔ 𝐹 ≠ 0 ))
2		fveq2 6668	. . . . . 6 ⊢ (𝑓 = 𝐹 → (coe1‘𝑓) = (coe1‘𝐹))
3		fveq2 6668	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝐷‘𝑓) = (𝐷‘𝐹))
4	2, 3	fveq12d 6675	. . . . 5 ⊢ (𝑓 = 𝐹 → ((coe1‘𝑓)‘(𝐷‘𝑓)) = ((coe1‘𝐹)‘(𝐷‘𝐹)))
5	4	eleq1d 2817	. . . 4 ⊢ (𝑓 = 𝐹 → (((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈 ↔ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))
6	1, 5	anbi12d 634	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈) ↔ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)))
7		uc1pval.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
8		uc1pval.b	. . . 4 ⊢ 𝐵 = (Base‘𝑃)
9		uc1pval.z	. . . 4 ⊢ 0 = (0g‘𝑃)
10		uc1pval.d	. . . 4 ⊢ 𝐷 = ( deg1 ‘𝑅)
11		uc1pval.c	. . . 4 ⊢ 𝐶 = (Unic1p‘𝑅)
12		uc1pval.u	. . . 4 ⊢ 𝑈 = (Unit‘𝑅)
13	7, 8, 9, 10, 11, 12	uc1pval 24884	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}
14	6, 13	elrab2 3588	. 2 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)))
15		3anass 1096	. 2 ⊢ ((𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈) ↔ (𝐹 ∈ 𝐵 ∧ (𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈)))
16	14, 15	bitr4i 281	1 ⊢ (𝐹 ∈ 𝐶 ↔ (𝐹 ∈ 𝐵 ∧ 𝐹 ≠ 0 ∧ ((coe1‘𝐹)‘(𝐷‘𝐹)) ∈ 𝑈))

Syntax hints:   ↔ wb 209   ∧ wa 399   ∧ w3a 1088   = wceq 1542   ∈ wcel 2113   ≠ wne 2934  ‘cfv 6333  Basecbs 16579  0_gc0g 16809  Unitcui 19504  Poly₁cpl1 20945  coe₁cco1 20946   deg₁ cdg1 24796  Unic_1pcuc1p 24871

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6291  df-fun 6335  df-fv 6341  df-slot 16583  df-base 16585  df-uc1p 24876

This theorem is referenced by:  uc1pcl  24888  uc1pn0  24890  uc1pldg  24893  mon1puc1p  24895  drnguc1p  24915