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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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