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Theorem mon1pn0 24747
Description: Monic polynomials are not zero. (Contributed by Stefan O'Rear, 28-Mar-2015.)
Hypotheses
Ref Expression
uc1pn0.p 𝑃 = (Poly1𝑅)
uc1pn0.z 0 = (0g𝑃)
mon1pn0.m 𝑀 = (Monic1p𝑅)
Assertion
Ref Expression
mon1pn0 (𝐹𝑀𝐹0 )

Proof of Theorem mon1pn0
StepHypRef Expression
1 uc1pn0.p . . 3 𝑃 = (Poly1𝑅)
2 eqid 2798 . . 3 (Base‘𝑃) = (Base‘𝑃)
3 uc1pn0.z . . 3 0 = (0g𝑃)
4 eqid 2798 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 mon1pn0.m . . 3 𝑀 = (Monic1p𝑅)
6 eqid 2798 . . 3 (1r𝑅) = (1r𝑅)
71, 2, 3, 4, 5, 6ismon1p 24743 . 2 (𝐹𝑀 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹0 ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) = (1r𝑅)))
87simp2bi 1143 1 (𝐹𝑀𝐹0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  wne 2987  cfv 6324  Basecbs 16475  0gc0g 16705  1rcur 19244  Poly1cpl1 20806  coe1cco1 20807   deg1 cdg1 24655  Monic1pcmn1 24726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-slot 16479  df-base 16481  df-mon1 24731
This theorem is referenced by:  mon1puc1p  24751  deg1submon1p  24753  mon1psubm  40148
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