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Theorem isuc1p 26180
Description: Being a unitic polynomial. (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
isuc1p (𝐹𝐶 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))

Proof of Theorem isuc1p
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 neeq1 3003 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6906 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6906 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6913 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eleq1d 2826 . . . 4 (𝑓 = 𝐹 → (((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈 ↔ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
61, 5anbi12d 632 . . 3 (𝑓 = 𝐹 → ((𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈) ↔ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
7 uc1pval.p . . . 4 𝑃 = (Poly1𝑅)
8 uc1pval.b . . . 4 𝐵 = (Base‘𝑃)
9 uc1pval.z . . . 4 0 = (0g𝑃)
10 uc1pval.d . . . 4 𝐷 = (deg1𝑅)
11 uc1pval.c . . . 4 𝐶 = (Unic1p𝑅)
12 uc1pval.u . . . 4 𝑈 = (Unit‘𝑅)
137, 8, 9, 10, 11, 12uc1pval 26179 . . 3 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
146, 13elrab2 3695 . 2 (𝐹𝐶 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
15 3anass 1095 . 2 ((𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈) ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
1614, 15bitr4i 278 1 (𝐹𝐶 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cfv 6561  Basecbs 17247  0gc0g 17484  Unitcui 20355  Poly1cpl1 22178  coe1cco1 22179  deg1cdg1 26093  Unic1pcuc1p 26166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-slot 17219  df-ndx 17231  df-base 17248  df-uc1p 26171
This theorem is referenced by:  uc1pcl  26183  uc1pn0  26185  uc1pldg  26188  mon1puc1p  26190  drnguc1p  26213
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