Theorem mon1pval 26310

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 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
3		uc1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
4	2, 3	eqtr4di 2815	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
5	4	fveq2d 6885	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
6		uc1pval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	eqtr4di 2815	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
8	4	fveq2d 6885	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
9		uc1pval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑃)
10	8, 9	eqtr4di 2815	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
11	10	neeq2d 3017	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 ))
12		fveq2 6881	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
13		uc1pval.d	. . . . . . . . . 10 ⊢ 𝐷 = (deg1‘𝑅)
14	12, 13	eqtr4di 2815	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷)
15	14	fveq1d 6883	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑓) = (𝐷‘𝑓))
16	15	fveq2d 6885	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓)))
17		fveq2 6881	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
18		mon1pval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
19	17, 18	eqtr4di 2815	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
20	16, 19	eqeq12d 2778	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟) ↔ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ))
21	11, 20	anbi12d 643	. . . . 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 3433	. . . 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 26299	. . . 4 ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
24	6	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
25	24	rabex 5308	. . . 4 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ∈ V
26	22, 23, 25	fvmpt 6989	. . 3 ⊢ (𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
27		fvprc 6873	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = ∅)
28		ssrab2 4033	. . . . . 6 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ 𝐵
29		fvprc 6873	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
30	3, 29	eqtrid 2809	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅)
31	30	fveq2d 6885	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32	6, 31	eqtrid 2809	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝐵 = (Base‘∅))
33		base0 17280	. . . . . . 7 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2815	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
35	28, 34	sseqtrid 3978	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅)
36		ss0 4358	. . . . 5 ⊢ ({𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅ → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} = ∅)
37	35, 36	syl 18	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} = ∅)
38	27, 37	eqtr4d 2800	. . 3 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
39	26, 38	pm2.61i 184	. 2 ⊢ (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}
40	1, 39	eqtri 2785	1 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  {crab 3415  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  ‘cfv 6536  Basecbs 17275  0_gc0g 17498  1_rcur 20269  Poly₁cpl1 22348  coe₁cco1 22349  deg₁cdg1 26222  Monic_1pcmn1 26294

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7415  df-om 7861  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-nn 12240  df-slot 17248  df-ndx 17260  df-base 17276  df-mon1 26299

This theorem is used by:  ismon1p  26311