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Theorem ismon1p 24122
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 3005 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6332 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6332 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6338 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eqeq1d 2773 . . . 4 (𝑓 = 𝐹 → (((coe1𝑓)‘(𝐷𝑓)) = 1 ↔ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))
61, 5anbi12d 616 . . 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 24121 . . 3 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
146, 13elrab2 3518 . 2 (𝐹𝑀 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
15 3anass 1080 . 2 ((𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ) ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
1614, 15bitr4i 267 1 (𝐹𝑀 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  wne 2943  cfv 6031  Basecbs 16064  0gc0g 16308  1rcur 18709  Poly1cpl1 19762  coe1cco1 19763   deg1 cdg1 24034  Monic1pcmn1 24105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-slot 16068  df-base 16070  df-mon1 24110
This theorem is referenced by:  mon1pcl  24124  mon1pn0  24126  mon1pldg  24129  uc1pmon1p  24131  ply1remlem  24142  mon1pid  38309  mon1psubm  38310
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