MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isuc1p Structured version   Visualization version   GIF version

Theorem isuc1p 23804
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 2852 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6148 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6148 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6154 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eleq1d 2683 . . . 4 (𝑓 = 𝐹 → (((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈 ↔ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
61, 5anbi12d 746 . . 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 23803 . . 3 𝐶 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) ∈ 𝑈)}
146, 13elrab2 3348 . 2 (𝐹𝐶 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
15 3anass 1040 . 2 ((𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈) ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈)))
1614, 15bitr4i 267 1 (𝐹𝐶 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) ∈ 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  cfv 5847  Basecbs 15781  0gc0g 16021  Unitcui 18560  Poly1cpl1 19466  coe1cco1 19467   deg1 cdg1 23718  Unic1pcuc1p 23790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-slot 15785  df-base 15786  df-uc1p 23795
This theorem is referenced by:  uc1pcl  23807  uc1pn0  23809  uc1pldg  23812  mon1puc1p  23814  drnguc1p  23834
  Copyright terms: Public domain W3C validator