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Theorem isuc1p 26195
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 3001 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6907 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6907 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6914 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eleq1d 2824 . . . 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 26194 . . 3 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
146, 13elrab2 3698 . 2 (𝐹𝐶 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
15 3anass 1094 . 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 1086   = wceq 1537  wcel 2106  wne 2938  cfv 6563  Basecbs 17245  0gc0g 17486  Unitcui 20372  Poly1cpl1 22194  coe1cco1 22195  deg1cdg1 26108  Unic1pcuc1p 26181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-1cn 11211  ax-addcl 11213
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-nn 12265  df-slot 17216  df-ndx 17228  df-base 17246  df-uc1p 26186
This theorem is referenced by:  uc1pcl  26198  uc1pn0  26200  uc1pldg  26203  mon1puc1p  26205  drnguc1p  26228
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