Theorem mon1pval 26099

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 6876	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
3		uc1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
4	2, 3	eqtr4di 2788	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
5	4	fveq2d 6880	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
6		uc1pval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	eqtr4di 2788	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
8	4	fveq2d 6880	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
9		uc1pval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑃)
10	8, 9	eqtr4di 2788	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
11	10	neeq2d 2992	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 ))
12		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
13		uc1pval.d	. . . . . . . . . 10 ⊢ 𝐷 = (deg1‘𝑅)
14	12, 13	eqtr4di 2788	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷)
15	14	fveq1d 6878	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑓) = (𝐷‘𝑓))
16	15	fveq2d 6880	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓)))
17		fveq2 6876	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
18		mon1pval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
19	17, 18	eqtr4di 2788	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
20	16, 19	eqeq12d 2751	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟) ↔ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 ))
21	11, 20	anbi12d 632	. . . . 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 3434	. . . 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 26088	. . . 4 ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
24	6	fvexi 6890	. . . . 5 ⊢ 𝐵 ∈ V
25	24	rabex 5309	. . . 4 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ∈ V
26	22, 23, 25	fvmpt 6986	. . 3 ⊢ (𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
27		fvprc 6868	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = ∅)
28		ssrab2 4055	. . . . . 6 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ 𝐵
29		fvprc 6868	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
30	3, 29	eqtrid 2782	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅)
31	30	fveq2d 6880	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32	6, 31	eqtrid 2782	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝐵 = (Base‘∅))
33		base0 17233	. . . . . . 7 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2788	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
35	28, 34	sseqtrid 4001	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅)
36		ss0 4377	. . . . 5 ⊢ ({𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅ → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} = ∅)
37	35, 36	syl 17	. . . 4 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} = ∅)
38	27, 37	eqtr4d 2773	. . 3 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
39	26, 38	pm2.61i 182	. 2 ⊢ (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}
40	1, 39	eqtri 2758	1 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  {crab 3415  Vcvv 3459   ⊆ wss 3926  ∅c0 4308  ‘cfv 6531  Basecbs 17228  0_gc0g 17453  1_rcur 20141  Poly₁cpl1 22112  coe₁cco1 22113  deg₁cdg1 26011  Monic_1pcmn1 26083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-1cn 11187  ax-addcl 11189

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-om 7862  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-nn 12241  df-slot 17201  df-ndx 17213  df-base 17229  df-mon1 26088

This theorem is referenced by:  ismon1p  26100