Theorem mon1pval 23901

Description: Value of the set of monic 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)
mon1pval.m	⊢ 𝑀 = (Monic1p‘𝑅)
mon1pval.o	⊢ 1 = (1r‘𝑅)

Assertion

Ref	Expression
mon1pval	⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}

Distinct variable groups:   𝐵,𝑓   𝐷,𝑓   1 ,𝑓   𝑅,𝑓   0 ,𝑓

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

Proof of Theorem mon1pval

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mon1pval.m	. 2 ⊢ 𝑀 = (Monic1p‘𝑅)
2		fveq2 6191	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
3		uc1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
4	2, 3	syl6eqr 2674	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
5	4	fveq2d 6195	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
6		uc1pval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	syl6eqr 2674	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
8	4	fveq2d 6195	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
9		uc1pval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑃)
10	8, 9	syl6eqr 2674	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
11	10	neeq2d 2854	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 ))
12		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = ( deg1 ‘𝑅))
13		uc1pval.d	. . . . . . . . . 10 ⊢ 𝐷 = ( deg1 ‘𝑅)
14	12, 13	syl6eqr 2674	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → ( deg1 ‘𝑟) = 𝐷)
15	14	fveq1d 6193	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (( deg1 ‘𝑟)‘𝑓) = (𝐷‘𝑓))
16	15	fveq2d 6195	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓)))
17		fveq2 6191	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
18		mon1pval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
19	17, 18	syl6eqr 2674	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
20	16, 19	eqeq12d 2637	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) = (1r‘𝑟) ↔ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ))
21	11, 20	anbi12d 747	. . . . 5 ⊢ (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) = (1r‘𝑟)) ↔ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )))
22	7, 21	rabeqbidv 3195	. . . 4 ⊢ (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) = (1r‘𝑟))} = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
23		df-mon1 23890	. . . 4 ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘(( deg1 ‘𝑟)‘𝑓)) = (1r‘𝑟))})
24		fvex 6201	. . . . . 6 ⊢ (Base‘𝑃) ∈ V
25	6, 24	eqeltri 2697	. . . . 5 ⊢ 𝐵 ∈ V
26	25	rabex 4813	. . . 4 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ∈ V
27	22, 23, 26	fvmpt 6282	. . 3 ⊢ (𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
28		fvprc 6185	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = ∅)
29		ssrab2 3687	. . . . . 6 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ 𝐵
30		fvprc 6185	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
31	3, 30	syl5eq 2668	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅)
32	31	fveq2d 6195	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
33	6, 32	syl5eq 2668	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝐵 = (Base‘∅))
34		base0 15912	. . . . . . 7 ⊢ ∅ = (Base‘∅)
35	33, 34	syl6eqr 2674	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
36	29, 35	syl5sseq 3653	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅)
37		ss0 3974	. . . . 5 ⊢ ({𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅ → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} = ∅)
38	36, 37	syl 17	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} = ∅)
39	28, 38	eqtr4d 2659	. . 3 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
40	27, 39	pm2.61i 176	. 2 ⊢ (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}
41	1, 40	eqtri 2644	1 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}

Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  {crab 2916  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ‘cfv 5888  Basecbs 15857  0_gc0g 16100  1_rcur 18501  Poly₁cpl1 19547  coe₁cco1 19548   deg₁ cdg1 23814  Monic_1pcmn1 23885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-slot 15861  df-base 15863  df-mon1 23890

This theorem is referenced by:  ismon1p  23902