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Theorem uc1pval 25527
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 6846 . . . . . . . 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 6850 . . . . . 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 6850 . . . . . . . 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 3001 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (𝑓 β‰  (0gβ€˜(Poly1β€˜π‘Ÿ)) ↔ 𝑓 β‰  0 ))
12 fveq2 6846 . . . . . . . . . 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 6848 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (( deg1 β€˜π‘Ÿ)β€˜π‘“) = (π·β€˜π‘“))
1615fveq2d 6850 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((coe1β€˜π‘“)β€˜(( deg1 β€˜π‘Ÿ)β€˜π‘“)) = ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)))
17 fveq2 6846 . . . . . . . 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 3423 . . . 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 25519 . . . 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 6860 . . . . 5 𝐡 ∈ V
2524rabex 5293 . . . 4 {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} ∈ V
2622, 23, 25fvmpt 6952 . . 3 (𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)})
27 fvprc 6838 . . . 4 (Β¬ 𝑅 ∈ V β†’ (Unic1pβ€˜π‘…) = βˆ…)
28 ssrab2 4041 . . . . . 6 {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† 𝐡
29 fvprc 6838 . . . . . . . . . 10 (Β¬ 𝑅 ∈ V β†’ (Poly1β€˜π‘…) = βˆ…)
303, 29eqtrid 2785 . . . . . . . . 9 (Β¬ 𝑅 ∈ V β†’ 𝑃 = βˆ…)
3130fveq2d 6850 . . . . . . . 8 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘ƒ) = (Baseβ€˜βˆ…))
32 base0 17096 . . . . . . . 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 4000 . . . . 5 (Β¬ 𝑅 ∈ V β†’ {𝑓 ∈ 𝐡 ∣ (𝑓 β‰  0 ∧ ((coe1β€˜π‘“)β€˜(π·β€˜π‘“)) ∈ π‘ˆ)} βŠ† βˆ…)
36 ss0 4362 . . . . 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 2940  {crab 3406  Vcvv 3447   βŠ† wss 3914  βˆ…c0 4286  β€˜cfv 6500  Basecbs 17091  0gc0g 17329  Unitcui 20076  Poly1cpl1 21571  coe1cco1 21572   deg1 cdg1 25439  Unic1pcuc1p 25514
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-1cn 11117  ax-addcl 11119
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 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-nn 12162  df-slot 17062  df-ndx 17074  df-base 17092  df-uc1p 25519
This theorem is referenced by:  isuc1p  25528
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