MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uc1pval Structured version   Visualization version   GIF version
Theorem uc1pval 25881
Description: Value of the set of unitic polynomials. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pval.p 𝑃 = (Poly1β€˜π‘…)
uc1pval.b 𝐡 = (Baseβ€˜π‘ƒ)
uc1pval.z 0 = (0gβ€˜π‘ƒ)
uc1pval.d 𝐷 = ( deg1 β€˜π‘…)
uc1pval.c 𝐢 = (Unic1pβ€˜π‘…)
uc1pval.u π‘ˆ = (Unitβ€˜π‘…)
Assertion
Ref Expression
uc1pval 𝐢 = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)}
Distinct variable groups:   𝐡,𝑓   𝐷,𝑓   𝑅,𝑓   π‘ˆ,𝑓   0 ,𝑓
Allowed substitution hints:   𝐢(𝑓)   𝑃(𝑓)
Proof of Theorem uc1pval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 uc1pval.c . 2 𝐢 = (Unic1pβ€˜π‘…)
2 fveq2 6891 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = (Poly1β€˜π‘…))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1β€˜π‘…)
42, 3eqtr4di 2790 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = 𝑃)
54fveq2d 6895 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(Poly1β€˜π‘Ÿ)) = (Baseβ€˜π‘ƒ))
6 uc1pval.b . . . . . 6 𝐡 = (Baseβ€˜π‘ƒ)
75, 6eqtr4di 2790 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(Poly1β€˜π‘Ÿ)) = 𝐡)
84fveq2d 6895 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (0gβ€˜(Poly1β€˜π‘Ÿ)) = (0gβ€˜π‘ƒ))
9 uc1pval.z . . . . . . . 8 0 = (0gβ€˜π‘ƒ)
108, 9eqtr4di 2790 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (0gβ€˜(Poly1β€˜π‘Ÿ)) = 0 )
1110neeq2d 3001 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (𝑓 β‰  (0gβ€˜(Poly1β€˜π‘Ÿ)) ↔ 𝑓 β‰  0 ))
12 fveq2 6891 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = ( deg1 β€˜π‘…))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1 β€˜π‘…)
1412, 13eqtr4di 2790 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = 𝐷)
1514fveq1d 6893 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘“) = (π·β€˜π‘“))
1615fveq2d 6895 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) = ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)))
17 fveq2 6891 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
18 uc1pval.u . . . . . . . 8 π‘ˆ = (Unitβ€˜π‘…)
1917, 18eqtr4di 2790 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
2016, 19eleq12d 2827 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) ∈ (Unitβ€˜π‘Ÿ) ↔ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ))
2111, 20anbi12d 631 . . . . 5 (π‘Ÿ = 𝑅 β†’ ((𝑓 β‰  (0gβ€˜(Poly1β€˜π‘Ÿ)) ∧ ((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) ∈ (Unitβ€˜π‘Ÿ)) ↔ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)))
227, 21rabeqbidv 3449 . . . 4 (π‘Ÿ = 𝑅 β†’ {𝑓 ∈ (Baseβ€˜(Poly1β€˜π‘Ÿ)) ∣ (𝑓 β‰  (0gβ€˜(Poly1β€˜π‘Ÿ)) ∧ ((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) ∈ (Unitβ€˜π‘Ÿ))} = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)})
23 df-uc1p 25873 . . . 4 Unic1p = (π‘Ÿ ∈ V ↦ {𝑓 ∈ (Baseβ€˜(Poly1β€˜π‘Ÿ)) ∣ (𝑓 β‰  (0gβ€˜(Poly1β€˜π‘Ÿ)) ∧ ((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) ∈ (Unitβ€˜π‘Ÿ))})
246fvexi 6905 . . . . 5 𝐡 ∈ V
2524rabex 5332 . . . 4 {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} ∈ V
2622, 23, 25fvmpt 6998 . . 3 (𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)})
27 fvprc 6883 . . . 4 (Β¬ 𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = βˆ…)
28 ssrab2 4077 . . . . . 6 {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† 𝐡
29 fvprc 6883 . . . . . . . . . 10 (Β¬ 𝑅 ∈ V β†’ (Poly1β€˜π‘…) = βˆ…)
303, 29eqtrid 2784 . . . . . . . . 9 (Β¬ 𝑅 ∈ V β†’ 𝑃 = βˆ…)
3130fveq2d 6895 . . . . . . . 8 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘ƒ) = (Baseβ€˜βˆ…))
32 base0 17153 . . . . . . . 8 βˆ… = (Baseβ€˜βˆ…)
3331, 32eqtr4di 2790 . . . . . . 7 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘ƒ) = βˆ…)
346, 33eqtrid 2784 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ 𝐡 = βˆ…)
3528, 34sseqtrid 4034 . . . . 5 (Β¬ 𝑅 ∈ V β†’ {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† βˆ…)
36 ss0 4398 . . . . 5 ({𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† βˆ… β†’ {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} = βˆ…)
3735, 36syl 17 . . . 4 (Β¬ 𝑅 ∈ V β†’ {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} = βˆ…)
3827, 37eqtr4d 2775 . . 3 (Β¬ 𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)})
3926, 38pm2.61i 182 . 2 (Unic1pβ€˜π‘…) = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)}
401, 39eqtri 2760 1 𝐢 = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  {crab 3432  Vcvv 3474   βŠ† wss 3948  βˆ…c0 4322  β€˜cfv 6543  Basecbs 17148  0gc0g 17389  Unitcui 20246  Poly1cpl1 21920  coe1cco1 21921   deg1 cdg1 25793  Unic1pcuc1p 25868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217  df-slot 17119  df-ndx 17131  df-base 17149  df-uc1p 25873
This theorem is referenced by:  isuc1p  25882
  Copyright terms: Public domain W3C validator