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Theorem mon1pn0 23710
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 2610 . . 3 (Base‘𝑃) = (Base‘𝑃)
3 uc1pn0.z . . 3 0 = (0g𝑃)
4 eqid 2610 . . 3 ( deg1𝑅) = ( deg1𝑅)
5 mon1pn0.m . . 3 𝑀 = (Monic1p𝑅)
6 eqid 2610 . . 3 (1r𝑅) = (1r𝑅)
71, 2, 3, 4, 5, 6ismon1p 23706 . 2 (𝐹𝑀 ↔ (𝐹 ∈ (Base‘𝑃) ∧ 𝐹0 ∧ ((coe1𝐹)‘(( deg1𝑅)‘𝐹)) = (1r𝑅)))
87simp2bi 1070 1 (𝐹𝑀𝐹0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = 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:  mon1puc1p  23714  deg1submon1p  23716  mon1psubm  36803
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