Theorem uc1pval 24740

Description: Value of the set of unitic polynomials. (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
uc1pval	⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}

Distinct variable groups:   𝐵,𝑓   𝐷,𝑓   𝑅,𝑓   𝑈,𝑓   0 ,𝑓

Allowed substitution hints:   𝐶(𝑓)   𝑃(𝑓)

Proof of Theorem uc1pval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uc1pval.c	. 2 ⊢ 𝐶 = (Unic1p‘𝑅)
2		fveq2 6645	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
3		uc1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
4	2, 3	eqtr4di 2851	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
5	4	fveq2d 6649	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
6		uc1pval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	eqtr4di 2851	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
8	4	fveq2d 6649	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
9		uc1pval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑃)
10	8, 9	eqtr4di 2851	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
11	10	neeq2d 3047	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 ))
12		fveq2 6645	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
13		uc1pval.d	. . . . . . . . . 10 ⊢ 𝐷 = ( deg1 ‘𝑅)
14	12, 13	eqtr4di 2851	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = 𝐷)
15	14	fveq1d 6647	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑓) = (𝐷‘𝑓))
16	15	fveq2d 6649	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓)))
17		fveq2 6645	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
18		uc1pval.u	. . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅)
19	17, 18	eqtr4di 2851	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
20	16, 19	eleq12d 2884	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟) ↔ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈))
21	11, 20	anbi12d 633	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟)) ↔ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)))
22	7, 21	rabeqbidv 3433	. . . 4 ⊢ (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))} = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)})
23		df-uc1p 24732	. . . 4 ⊢ Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
24	6	fvexi 6659	. . . . 5 ⊢ 𝐵 ∈ V
25	24	rabex 5199	. . . 4 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ∈ V
26	22, 23, 25	fvmpt 6745	. . 3 ⊢ (𝑅 ∈ V → (Unic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)})
27		fvprc 6638	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Unic1p‘𝑅) = ∅)
28		ssrab2 4007	. . . . . 6 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ⊆ 𝐵
29		fvprc 6638	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
30	3, 29	syl5eq 2845	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅)
31	30	fveq2d 6649	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32		base0 16528	. . . . . . . 8 ⊢ ∅ = (Base‘∅)
33	31, 32	eqtr4di 2851	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = ∅)
34	6, 33	syl5eq 2845	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
35	28, 34	sseqtrid 3967	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ⊆ ∅)
36		ss0 4306	. . . . 5 ⊢ ({𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} ⊆ ∅ → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} = ∅)
37	35, 36	syl 17	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)} = ∅)
38	27, 37	eqtr4d 2836	. . 3 ⊢ (¬ 𝑅 ∈ V → (Unic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)})
39	26, 38	pm2.61i 185	. 2 ⊢ (Unic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}
40	1, 39	eqtri 2821	1 ⊢ 𝐶 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) ∈ 𝑈)}

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  ‘cfv 6324  Basecbs 16475  0_gc0g 16705  Unitcui 19385  Poly₁cpl1 20806  coe₁cco1 20807   deg₁ cdg1 24655  Unic_1pcuc1p 24727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-slot 16479  df-base 16481  df-uc1p 24732

This theorem is referenced by:  isuc1p  24741