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Theorem uc1pval 24298
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 6433 . . . . . . . 8 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
3 uc1pval.p . . . . . . . 8 𝑃 = (Poly1𝑅)
42, 3syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (Poly1𝑟) = 𝑃)
54fveq2d 6437 . . . . . 6 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = (Base‘𝑃))
6 uc1pval.b . . . . . 6 𝐵 = (Base‘𝑃)
75, 6syl6eqr 2879 . . . . 5 (𝑟 = 𝑅 → (Base‘(Poly1𝑟)) = 𝐵)
84fveq2d 6437 . . . . . . . 8 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = (0g𝑃))
9 uc1pval.z . . . . . . . 8 0 = (0g𝑃)
108, 9syl6eqr 2879 . . . . . . 7 (𝑟 = 𝑅 → (0g‘(Poly1𝑟)) = 0 )
1110neeq2d 3059 . . . . . 6 (𝑟 = 𝑅 → (𝑓 ≠ (0g‘(Poly1𝑟)) ↔ 𝑓0 ))
12 fveq2 6433 . . . . . . . . . 10 (𝑟 = 𝑅 → ( deg1𝑟) = ( deg1𝑅))
13 uc1pval.d . . . . . . . . . 10 𝐷 = ( deg1𝑅)
1412, 13syl6eqr 2879 . . . . . . . . 9 (𝑟 = 𝑅 → ( deg1𝑟) = 𝐷)
1514fveq1d 6435 . . . . . . . 8 (𝑟 = 𝑅 → (( deg1𝑟)‘𝑓) = (𝐷𝑓))
1615fveq2d 6437 . . . . . . 7 (𝑟 = 𝑅 → ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) = ((coe1𝑓)‘(𝐷𝑓)))
17 fveq2 6433 . . . . . . . 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 2900 . . . . . 6 (𝑟 = 𝑅 → (((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟) ↔ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈))
2111, 20anbi12d 626 . . . . 5 (𝑟 = 𝑅 → ((𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟)) ↔ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)))
227, 21rabeqbidv 3408 . . . 4 (𝑟 = 𝑅 → {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))} = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
23 df-uc1p 24290 . . . 4 Unic1p = (𝑟 ∈ V ↦ {𝑓 ∈ (Base‘(Poly1𝑟)) ∣ (𝑓 ≠ (0g‘(Poly1𝑟)) ∧ ((coe1𝑓)‘(( deg1𝑟)‘𝑓)) ∈ (Unit‘𝑟))})
246fvexi 6447 . . . . 5 𝐵 ∈ V
2524rabex 5037 . . . 4 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ∈ V
2622, 23, 25fvmpt 6529 . . 3 (𝑅 ∈ V → (Unic1p𝑅) = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)})
27 fvprc 6426 . . . 4 𝑅 ∈ V → (Unic1p𝑅) = ∅)
28 ssrab2 3912 . . . . . 6 {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ 𝐵
29 fvprc 6426 . . . . . . . . . 10 𝑅 ∈ V → (Poly1𝑅) = ∅)
303, 29syl5eq 2873 . . . . . . . . 9 𝑅 ∈ V → 𝑃 = ∅)
3130fveq2d 6437 . . . . . . . 8 𝑅 ∈ V → (Base‘𝑃) = (Base‘∅))
32 base0 16275 . . . . . . . 8 ∅ = (Base‘∅)
3331, 32syl6eqr 2879 . . . . . . 7 𝑅 ∈ V → (Base‘𝑃) = ∅)
346, 33syl5eq 2873 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
3528, 34syl5sseq 3878 . . . . 5 𝑅 ∈ V → {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)} ⊆ ∅)
36 ss0 4199 . . . . 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 177 . 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 386   = wceq 1658  wcel 2166  wne 2999  {crab 3121  Vcvv 3414  wss 3798  c0 4144  cfv 6123  Basecbs 16222  0gc0g 16453  Unitcui 18993  Poly1cpl1 19907  coe1cco1 19908   deg1 cdg1 24213  Unic1pcuc1p 24285
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-iota 6086  df-fun 6125  df-fv 6131  df-slot 16226  df-base 16228  df-uc1p 24290
This theorem is referenced by:  isuc1p  24299
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