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Theorem isuc1p 25305
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 3006 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6774 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6774 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6781 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eleq1d 2823 . . . 4 (𝑓 = 𝐹 → (((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈 ↔ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
61, 5anbi12d 631 . . 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 25304 . . 3 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
146, 13elrab2 3627 . 2 (𝐹𝐶 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
15 3anass 1094 . 2 ((𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈) ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
1614, 15bitr4i 277 1 (𝐹𝐶 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wne 2943  cfv 6433  Basecbs 16912  0gc0g 17150  Unitcui 19881  Poly1cpl1 21348  coe1cco1 21349   deg1 cdg1 25216  Unic1pcuc1p 25291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-1cn 10929  ax-addcl 10931
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-nn 11974  df-slot 16883  df-ndx 16895  df-base 16913  df-uc1p 25296
This theorem is referenced by:  uc1pcl  25308  uc1pn0  25310  uc1pldg  25313  mon1puc1p  25315  drnguc1p  25335
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