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Theorem mon1pval 25304
Description: Value of the set of monic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p 𝑃 = (Poly1𝑅)
uc1pval.b 𝐵 = (Base‘𝑃)
uc1pval.z 0 = (0g𝑃)
uc1pval.d 𝐷 = ( deg1𝑅)
mon1pval.m 𝑀 = (Monic1p𝑅)
mon1pval.o 1 = (1r𝑅)
Assertion
Ref Expression
mon1pval 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 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.
StepHypRef Expression
1 mon1pval.m . 2 𝑀 = (Monic1p𝑅)
2 fveq2 6771 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1𝑅)
42, 3eqtr4di 2798 . . . . . . 7 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
54fveq2d 6775 . . . . . 6 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
6 uc1pval.b . . . . . 6 𝐵 = (Base‘𝑃)
75, 6eqtr4di 2798 . . . . 5 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
84fveq2d 6775 . . . . . . . 8 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
9 uc1pval.z . . . . . . . 8 0 = (0g𝑃)
108, 9eqtr4di 2798 . . . . . . 7 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1110neeq2d 3006 . . . . . 6 (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1𝑟)) ↔ 𝑓0 ))
12 fveq2 6771 . . . . . . . . . 10 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1𝑅)
1412, 13eqtr4di 2798 . . . . . . . . 9 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1514fveq1d 6773 . . . . . . . 8 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑓) = (𝐷𝑓))
1615fveq2d 6775 . . . . . . 7 (𝑟 = 𝑅 → ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = ((coe1𝑓)‘(𝐷𝑓)))
17 fveq2 6771 . . . . . . . 8 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
18 mon1pval.o . . . . . . . 8 1 = (1r𝑅)
1917, 18eqtr4di 2798 . . . . . . 7 (𝑟 = 𝑅 → (1r𝑟) = 1 )
2016, 19eqeq12d 2756 . . . . . 6 (𝑟 = 𝑅 → (((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟) ↔ ((coe1𝑓)‘(𝐷𝑓)) = 1 ))
2111, 20anbi12d 631 . . . . 5 (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟)) ↔ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )))
227, 21rabeqbidv 3419 . . . 4 (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))} = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
23 df-mon1 25293 . . . 4 Monic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = (1r𝑟))})
246fvexi 6785 . . . . 5 𝐵 ∈ V
2524rabex 5260 . . . 4 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ∈ V
2622, 23, 25fvmpt 6872 . . 3 (𝑅 ∈ V → (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
27 fvprc 6763 . . . 4 𝑅 ∈ V → (Monic1p𝑅) = ∅)
28 ssrab2 4018 . . . . . 6 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ 𝐵
29 fvprc 6763 . . . . . . . . . 10 𝑅 ∈ V → (Poly1𝑅) = ∅)
303, 29eqtrid 2792 . . . . . . . . 9 𝑅 ∈ V → 𝑃 = ∅)
3130fveq2d 6775 . . . . . . . 8 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
326, 31eqtrid 2792 . . . . . . 7 𝑅 ∈ V → 𝐵 = (Base‘∅))
33 base0 16915 . . . . . . 7 ∅ = (Base‘∅)
3432, 33eqtr4di 2798 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
3528, 34sseqtrid 3978 . . . . 5 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ ∅)
36 ss0 4338 . . . . 5 ({𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} ⊆ ∅ → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} = ∅)
3735, 36syl 17 . . . 4 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )} = ∅)
3827, 37eqtr4d 2783 . . 3 𝑅 ∈ V → (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )})
3926, 38pm2.61i 182 . 2 (Monic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
401, 39eqtri 2768 1 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1542  wcel 2110  wne 2945  {crab 3070  Vcvv 3431  wss 3892  c0 4262  cfv 6432  Basecbs 16910  0gc0g 17148  1rcur 19735  Poly1cpl1 21346  coe1cco1 21347   deg1 cdg1 25214  Monic1pcmn1 25288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-1cn 10930  ax-addcl 10932
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-om 7707  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-nn 11974  df-slot 16881  df-ndx 16893  df-base 16911  df-mon1 25293
This theorem is referenced by:  ismon1p  25305
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