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Theorem ismon1p 23706
 Description: Being a monic 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)
mon1pval.m 𝑀 = (Monic1p𝑅)
mon1pval.o 1 = (1r𝑅)
Assertion
Ref Expression
ismon1p (𝐹𝑀 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))

Proof of Theorem ismon1p
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 neeq1 2844 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6103 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6103 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6109 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eqeq1d 2612 . . . 4 (𝑓 = 𝐹 → (((coe1𝑓)‘(𝐷𝑓)) = 1 ↔ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))
61, 5anbi12d 743 . . 3 (𝑓 = 𝐹 → ((𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 ) ↔ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
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 mon1pval.m . . . 4 𝑀 = (Monic1p𝑅)
12 mon1pval.o . . . 4 1 = (1r𝑅)
137, 8, 9, 10, 11, 12mon1pval 23705 . . 3 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
146, 13elrab2 3333 . 2 (𝐹𝑀 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
15 3anass 1035 . 2 ((𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ) ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
1614, 15bitr4i 266 1 (𝐹𝑀 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ‘cfv 5804  Basecbs 15695  0gc0g 15923  1rcur 18324  Poly1cpl1 19368  coe1cco1 19369   deg1 cdg1 23618  Monic1pcmn1 23689 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699  df-base 15700  df-mon1 23694 This theorem is referenced by:  mon1pcl  23708  mon1pn0  23710  mon1pldg  23713  uc1pmon1p  23715  ply1remlem  23726  mon1pid  36802  mon1psubm  36803
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