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Theorem uc1pval 26167
Description: Value of the set of unitic 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)
uc1pval.c 𝐶 = (Unic1p𝑅)
uc1pval.u 𝑈 = (Unit‘𝑅)
Assertion
Ref Expression
uc1pval 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
Distinct variable groups:   𝐵,𝑓   𝐷,𝑓   𝑅,𝑓   𝑈,𝑓   0 ,𝑓
Allowed substitution hints:   𝐶(𝑓)   𝑃(𝑓)

Proof of Theorem uc1pval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 uc1pval.c . 2 𝐶 = (Unic1p𝑅)
2 fveq2 6901 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1𝑅)
42, 3eqtr4di 2784 . . . . . . 7 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
54fveq2d 6905 . . . . . 6 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
6 uc1pval.b . . . . . 6 𝐵 = (Base‘𝑃)
75, 6eqtr4di 2784 . . . . 5 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
84fveq2d 6905 . . . . . . . 8 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
9 uc1pval.z . . . . . . . 8 0 = (0g𝑃)
108, 9eqtr4di 2784 . . . . . . 7 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1110neeq2d 2991 . . . . . 6 (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1𝑟)) ↔ 𝑓0 ))
12 fveq2 6901 . . . . . . . . . 10 (𝑟 = 𝑅 → (deg1𝑟) = (deg1𝑅))
13 uc1pval.d . . . . . . . . . 10 𝐷 = (deg1𝑅)
1412, 13eqtr4di 2784 . . . . . . . . 9 (𝑟 = 𝑅 → (deg1𝑟) = 𝐷)
1514fveq1d 6903 . . . . . . . 8 (𝑟 = 𝑅 → ((deg1𝑟)‘𝑓) = (𝐷𝑓))
1615fveq2d 6905 . . . . . . 7 (𝑟 = 𝑅 → ((coe1𝑓)‘((deg1𝑟)‘𝑓)) = ((coe1𝑓)‘(𝐷𝑓)))
17 fveq2 6901 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
18 uc1pval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
1917, 18eqtr4di 2784 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
2016, 19eleq12d 2820 . . . . . 6 (𝑟 = 𝑅 → (((coe1𝑓)‘((deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟) ↔ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈))
2111, 20anbi12d 630 . . . . 5 (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘((deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟)) ↔ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)))
227, 21rabeqbidv 3437 . . . 4 (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘((deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))} = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
23 df-uc1p 26159 . . . 4 Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘((deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
246fvexi 6915 . . . . 5 𝐵 ∈ V
2524rabex 5339 . . . 4 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ∈ V
2622, 23, 25fvmpt 7009 . . 3 (𝑅 ∈ V → (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
27 fvprc 6893 . . . 4 𝑅 ∈ V → (Unic1p𝑅) = ∅)
28 ssrab2 4076 . . . . . 6 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ 𝐵
29 fvprc 6893 . . . . . . . . . 10 𝑅 ∈ V → (Poly1𝑅) = ∅)
303, 29eqtrid 2778 . . . . . . . . 9 𝑅 ∈ V → 𝑃 = ∅)
3130fveq2d 6905 . . . . . . . 8 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32 base0 17218 . . . . . . . 8 ∅ = (Base‘∅)
3331, 32eqtr4di 2784 . . . . . . 7 𝑅 ∈ V → (Base‘𝑃) = ∅)
346, 33eqtrid 2778 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
3528, 34sseqtrid 4032 . . . . 5 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ ∅)
36 ss0 4403 . . . . 5 ({𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ ∅ → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} = ∅)
3735, 36syl 17 . . . 4 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} = ∅)
3827, 37eqtr4d 2769 . . 3 𝑅 ∈ V → (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
3926, 38pm2.61i 182 . 2 (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
401, 39eqtri 2754 1 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394   = wceq 1534  wcel 2099  wne 2930  {crab 3419  Vcvv 3462  wss 3947  c0 4325  cfv 6554  Basecbs 17213  0gc0g 17454  Unitcui 20337  Poly1cpl1 22166  coe1cco1 22167  deg1cdg1 26078  Unic1pcuc1p 26154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-1cn 11216  ax-addcl 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-om 7877  df-2nd 8004  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-nn 12265  df-slot 17184  df-ndx 17196  df-base 17214  df-uc1p 26159
This theorem is referenced by:  isuc1p  26168
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