Theorem mon1pval 26256

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 6871	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = (Poly1‘𝑅))
3		uc1pval.p	. . . . . . . 8 ⊢ 𝑃 = (Poly1‘𝑅)
4	2, 3	eqtr4di 2818	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Poly1‘𝑟) = 𝑃)
5	4	fveq2d 6875	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = (Base‘𝑃))
6		uc1pval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑃)
7	5, 6	eqtr4di 2818	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘(Poly1‘𝑟)) = 𝐵)
8	4	fveq2d 6875	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = (0g‘𝑃))
9		uc1pval.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑃)
10	8, 9	eqtr4di 2818	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (0g‘(Poly1‘𝑟)) = 0 )
11	10	neeq2d 3020	. . . . . 6 ⊢ (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1‘𝑟)) ↔ 𝑓 ≠ 0 ))
12		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = (deg1‘𝑅))
13		uc1pval.d	. . . . . . . . . 10 ⊢ 𝐷 = (deg1‘𝑅)
14	12, 13	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (deg1‘𝑟) = 𝐷)
15	14	fveq1d 6873	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → ((deg1‘𝑟)‘𝑓) = (𝐷‘𝑓))
16	15	fveq2d 6875	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = ((coe1‘𝑓)‘(𝐷‘𝑓)))
17		fveq2 6871	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = (1r‘𝑅))
18		mon1pval.o	. . . . . . . 8 ⊢ 1 = (1r‘𝑅)
19	17, 18	eqtr4di 2818	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (1r‘𝑟) = 1 )
20	16, 19	eqeq12d 2781	. . . . . 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 3435	. . . 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 26245	. . . 4 ⊢ Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1‘𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1‘𝑟)) ∧ ((coe1‘𝑓)‘((deg1‘𝑟)‘𝑓)) = (1r‘𝑟))})
24	6	fvexi 6885	. . . . 5 ⊢ 𝐵 ∈ V
25	24	rabex 5299	. . . 4 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ∈ V
26	22, 23, 25	fvmpt 6979	. . 3 ⊢ (𝑅 ∈ V → (Monic1p‘𝑅) = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )})
27		fvprc 6863	. . . 4 ⊢ (¬ 𝑅 ∈ V → (Monic1p‘𝑅) = ∅)
28		ssrab2 4036	. . . . . 6 ⊢ {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ 𝐵
29		fvprc 6863	. . . . . . . . . 10 ⊢ (¬ 𝑅 ∈ V → (Poly1‘𝑅) = ∅)
30	3, 29	eqtrid 2812	. . . . . . . . 9 ⊢ (¬ 𝑅 ∈ V → 𝑃 = ∅)
31	30	fveq2d 6875	. . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32	6, 31	eqtrid 2812	. . . . . . 7 ⊢ (¬ 𝑅 ∈ V → 𝐵 = (Base‘∅))
33		base0 17262	. . . . . . 7 ⊢ ∅ = (Base‘∅)
34	32, 33	eqtr4di 2818	. . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐵 = ∅)
35	28, 34	sseqtrid 3981	. . . . 5 ⊢ (¬ 𝑅 ∈ V → {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )} ⊆ ∅)
36		ss0 4359	. . . . 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 2803	. . 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 2788	1 ⊢ 𝑀 = {𝑓 ∈ 𝐵 ∣ (𝑓 ≠ 0 ∧ ((coe1‘𝑓)‘(𝐷‘𝑓)) = 1 )}

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  {crab 3417  Vcvv 3457   ⊆ wss 3907  ∅c0 4288  ‘cfv 6525  Basecbs 17257  0_gc0g 17480  1_rcur 20251  Poly₁cpl1 22294  coe₁cco1 22295  deg₁cdg1 26168  Monic_1pcmn1 26240

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-1cn 11146  ax-addcl 11148

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-nn 12222  df-slot 17230  df-ndx 17242  df-base 17258  df-mon1 26245

This theorem is referenced by:  ismon1p  26257