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Theorem uc1pval 24648
 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 6667 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1𝑅)
42, 3syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
54fveq2d 6671 . . . . . 6 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
6 uc1pval.b . . . . . 6 𝐵 = (Base‘𝑃)
75, 6syl6eqr 2879 . . . . 5 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
84fveq2d 6671 . . . . . . . 8 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
9 uc1pval.z . . . . . . . 8 0 = (0g𝑃)
108, 9syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1110neeq2d 3081 . . . . . 6 (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1𝑟)) ↔ 𝑓0 ))
12 fveq2 6667 . . . . . . . . . 10 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1𝑅)
1412, 13syl6eqr 2879 . . . . . . . . 9 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1514fveq1d 6669 . . . . . . . 8 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑓) = (𝐷𝑓))
1615fveq2d 6671 . . . . . . 7 (𝑟 = 𝑅 → ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = ((coe1𝑓)‘(𝐷𝑓)))
17 fveq2 6667 . . . . . . . 8 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
18 uc1pval.u . . . . . . . 8 𝑈 = (Unit‘𝑅)
1917, 18syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
2016, 19eleq12d 2912 . . . . . 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 3491 . . . 4 (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))} = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
23 df-uc1p 24640 . . . 4 Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
246fvexi 6681 . . . . 5 𝐵 ∈ V
2524rabex 5232 . . . 4 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ∈ V
2622, 23, 25fvmpt 6765 . . 3 (𝑅 ∈ V → (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
27 fvprc 6660 . . . 4 𝑅 ∈ V → (Unic1p𝑅) = ∅)
28 ssrab2 4060 . . . . . 6 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ 𝐵
29 fvprc 6660 . . . . . . . . . 10 𝑅 ∈ V → (Poly1𝑅) = ∅)
303, 29syl5eq 2873 . . . . . . . . 9 𝑅 ∈ V → 𝑃 = ∅)
3130fveq2d 6671 . . . . . . . 8 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32 base0 16526 . . . . . . . 8 ∅ = (Base‘∅)
3331, 32syl6eqr 2879 . . . . . . 7 𝑅 ∈ V → (Base‘𝑃) = ∅)
346, 33syl5eq 2873 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
3528, 34sseqtrid 4023 . . . . 5 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ ∅)
36 ss0 4356 . . . . 5 ({𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ ∅ → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} = ∅)
3735, 36syl 17 . . . 4 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} = ∅)
3827, 37eqtr4d 2864 . . 3 𝑅 ∈ V → (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
3926, 38pm2.61i 183 . 2 (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
401, 39eqtri 2849 1 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1530   ∈ wcel 2107   ≠ wne 3021  {crab 3147  Vcvv 3500   ⊆ wss 3940  ∅c0 4295  ‘cfv 6352  Basecbs 16473  0gc0g 16703  Unitcui 19309  Poly1cpl1 20262  coe1cco1 20263   deg1 cdg1 24563  Unic1pcuc1p 24635 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fv 6360  df-slot 16477  df-base 16479  df-uc1p 24640 This theorem is referenced by:  isuc1p  24649
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