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Theorem ismon1p 25040
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 3003 . . . 4 (𝑓 = 𝐹 → (𝑓0𝐹0 ))
2 fveq2 6717 . . . . . 6 (𝑓 = 𝐹 → (coe1𝑓) = (coe1𝐹))
3 fveq2 6717 . . . . . 6 (𝑓 = 𝐹 → (𝐷𝑓) = (𝐷𝐹))
42, 3fveq12d 6724 . . . . 5 (𝑓 = 𝐹 → ((coe1𝑓)‘(𝐷𝑓)) = ((coe1𝐹)‘(𝐷𝐹)))
54eqeq1d 2739 . . . 4 (𝑓 = 𝐹 → (((coe1𝑓)‘(𝐷𝑓)) = 1 ↔ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))
61, 5anbi12d 634 . . 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 25039 . . 3 𝑀 = {𝑓𝐵 ∣ (𝑓0 ∧ ((coe1𝑓)‘(𝐷𝑓)) = 1 )}
146, 13elrab2 3605 . 2 (𝐹𝑀 ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
15 3anass 1097 . 2 ((𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ) ↔ (𝐹𝐵 ∧ (𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 )))
1614, 15bitr4i 281 1 (𝐹𝑀 ↔ (𝐹𝐵𝐹0 ∧ ((coe1𝐹)‘(𝐷𝐹)) = 1 ))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wne 2940  cfv 6380  Basecbs 16760  0gc0g 16944  1rcur 19516  Poly1cpl1 21098  coe1cco1 21099   deg1 cdg1 24949  Monic1pcmn1 25023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-1cn 10787  ax-addcl 10789
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-om 7645  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-nn 11831  df-slot 16735  df-ndx 16745  df-base 16761  df-mon1 25028
This theorem is referenced by:  mon1pcl  25042  mon1pn0  25044  mon1pldg  25047  uc1pmon1p  25049  ply1remlem  25060  mon1pid  40733  mon1psubm  40734
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