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Theorem uc1pval 25657
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 6892 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = (Poly1β€˜π‘…))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1β€˜π‘…)
42, 3eqtr4di 2791 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Poly1β€˜π‘Ÿ) = 𝑃)
54fveq2d 6896 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(Poly1β€˜π‘Ÿ)) = (Baseβ€˜π‘ƒ))
6 uc1pval.b . . . . . 6 𝐡 = (Baseβ€˜π‘ƒ)
75, 6eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜(Poly1β€˜π‘Ÿ)) = 𝐡)
84fveq2d 6896 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (0gβ€˜(Poly1β€˜π‘Ÿ)) = (0gβ€˜π‘ƒ))
9 uc1pval.z . . . . . . . 8 0 = (0gβ€˜π‘ƒ)
108, 9eqtr4di 2791 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (0gβ€˜(Poly1β€˜π‘Ÿ)) = 0 )
1110neeq2d 3002 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (𝑓 β‰  (0gβ€˜(Poly1β€˜π‘Ÿ)) ↔ 𝑓 β‰  0 ))
12 fveq2 6892 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = ( deg1 β€˜π‘…))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1 β€˜π‘…)
1412, 13eqtr4di 2791 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ ( deg1 β€˜π‘Ÿ) = 𝐷)
1514fveq1d 6894 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘“) = (π·β€˜π‘“))
1615fveq2d 6896 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) = ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)))
17 fveq2 6892 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
18 uc1pval.u . . . . . . . 8 π‘ˆ = (Unitβ€˜π‘…)
1917, 18eqtr4di 2791 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
2016, 19eleq12d 2828 . . . . . 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 632 . . . . 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 3450 . . . 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 25649 . . . 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 6906 . . . . 5 𝐡 ∈ V
2524rabex 5333 . . . 4 {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} ∈ V
2622, 23, 25fvmpt 6999 . . 3 (𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)})
27 fvprc 6884 . . . 4 (Β¬ 𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = βˆ…)
28 ssrab2 4078 . . . . . 6 {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† 𝐡
29 fvprc 6884 . . . . . . . . . 10 (Β¬ 𝑅 ∈ V β†’ (Poly1β€˜π‘…) = βˆ…)
303, 29eqtrid 2785 . . . . . . . . 9 (Β¬ 𝑅 ∈ V β†’ 𝑃 = βˆ…)
3130fveq2d 6896 . . . . . . . 8 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘ƒ) = (Baseβ€˜βˆ…))
32 base0 17149 . . . . . . . 8 βˆ… = (Baseβ€˜βˆ…)
3331, 32eqtr4di 2791 . . . . . . 7 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘ƒ) = βˆ…)
346, 33eqtrid 2785 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ 𝐡 = βˆ…)
3528, 34sseqtrid 4035 . . . . 5 (Β¬ 𝑅 ∈ V β†’ {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† βˆ…)
36 ss0 4399 . . . . 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 2776 . . 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 2761 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 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  {crab 3433  Vcvv 3475   βŠ† wss 3949  βˆ…c0 4323  β€˜cfv 6544  Basecbs 17144  0gc0g 17385  Unitcui 20169  Poly1cpl1 21701  coe1cco1 21702   deg1 cdg1 25569  Unic1pcuc1p 25644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-slot 17115  df-ndx 17127  df-base 17145  df-uc1p 25649
This theorem is referenced by:  isuc1p  25658
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