Theorem mon1pval 26515

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 6916	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
3		uc1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
4	2, 3	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
5	4	fveq2d 6920	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
6		uc1pval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	eqtr4di 2816	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
8	4	fveq2d 6920	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
9		uc1pval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑃)
10	8, 9	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
11	10	neeq2d 3019	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 ))
12		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
13		uc1pval.d	. . . . . . . . . 10 ⊢ 𝐷 = (deg1‘𝑅)
14	12, 13	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷)
15	14	fveq1d 6918	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑓) = (𝐷‘𝑓))
16	15	fveq2d 6920	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓)))
17		fveq2 6916	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
18		mon1pval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
19	17, 18	eqtr4di 2816	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
20	16, 19	eqeq12d 2779	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟) ↔ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ))
21	11, 20	anbi12d 644	. . . . 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 26504	. . . 4 ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
24	6	fvexi 6930	. . . . 5 ⊢ 𝐵 ∈ V
25	24	rabex 5307	. . . 4 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ∈ V
26	22, 23, 25	fvmpt 7024	. . 3 ⊢ (𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
27		fvprc 6908	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = ∅)
28		ssrab2 4008	. . . . . 6 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ 𝐵
29		fvprc 6908	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
30	3, 29	eqtrid 2810	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅)
31	30	fveq2d 6920	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32	6, 31	eqtrid 2810	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝐵 = (Base‘∅))
33		base0 17436	. . . . . . 7 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2816	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
35	28, 34	sseqtrid 3955	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅)
36		ss0 4355	. . . . 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 2801	. . 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 2786	1 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  {crab 3416  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  ‘cfv 6566  Basecbs 17431  0_gc0g 17660  1_rcur 20458  Poly₁cpl1 22550  coe₁cco1 22551  deg₁cdg1 26427  Monic_1pcmn1 26499

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-1cn 11297  ax-addcl 11299

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-nn 12376  df-slot 17404  df-ndx 17416  df-base 17432  df-mon1 26504

This theorem is used by:  ismon1p  26516