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Theorem drnguc1p 19609
Description: Over a division ring, all nonzero polynomials are unitic. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypotheses
Ref Expression
drnguc1p.p  |-  P  =  (Poly1 `  R )
drnguc1p.b  |-  B  =  ( Base `  P
)
drnguc1p.z  |-  .0.  =  ( 0g `  P )
drnguc1p.c  |-  C  =  (Unic1p `  R )
Assertion
Ref Expression
drnguc1p  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  F  e.  C )

Proof of Theorem drnguc1p
StepHypRef Expression
1 simp2 956 . 2  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  F  e.  B )
2 simp3 957 . 2  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  F  =/=  .0.  )
3 eqid 2316 . . . . . 6  |-  (coe1 `  F
)  =  (coe1 `  F
)
4 drnguc1p.b . . . . . 6  |-  B  =  ( Base `  P
)
5 drnguc1p.p . . . . . 6  |-  P  =  (Poly1 `  R )
6 eqid 2316 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
73, 4, 5, 6coe1f 16341 . . . . 5  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
873ad2ant2 977 . . . 4  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
9 drngrng 15568 . . . . 5  |-  ( R  e.  DivRing  ->  R  e.  Ring )
10 eqid 2316 . . . . . 6  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
11 drnguc1p.z . . . . . 6  |-  .0.  =  ( 0g `  P )
1210, 5, 11, 4deg1nn0cl 19527 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
( deg1  `
 R ) `  F )  e.  NN0 )
139, 12syl3an1 1215 . . . 4  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
( deg1  `
 R ) `  F )  e.  NN0 )
14 ffvelrn 5701 . . . 4  |-  ( ( (coe1 `  F ) : NN0 --> ( Base `  R
)  /\  ( ( deg1  `  R ) `  F
)  e.  NN0 )  ->  ( (coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  e.  ( Base `  R
) )
158, 13, 14syl2anc 642 . . 3  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
(coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  e.  ( Base `  R
) )
16 eqid 2316 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
1710, 5, 11, 4, 16, 3deg1ldg 19531 . . . 4  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
(coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  =/=  ( 0g `  R
) )
189, 17syl3an1 1215 . . 3  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
(coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  =/=  ( 0g `  R
) )
19 eqid 2316 . . . . 5  |-  (Unit `  R )  =  (Unit `  R )
206, 19, 16drngunit 15566 . . . 4  |-  ( R  e.  DivRing  ->  ( ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  e.  (Unit `  R )  <->  ( ( (coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  e.  ( Base `  R
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  =/=  ( 0g `  R
) ) ) )
21203ad2ant1 976 . . 3  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
( (coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  e.  (Unit `  R )  <->  ( ( (coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  e.  ( Base `  R
)  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  =/=  ( 0g `  R
) ) ) )
2215, 18, 21mpbir2and 888 . 2  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  (
(coe1 `  F ) `  ( ( deg1  `  R ) `  F ) )  e.  (Unit `  R )
)
23 drnguc1p.c . . 3  |-  C  =  (Unic1p `  R )
245, 4, 11, 10, 23, 19isuc1p 19579 . 2  |-  ( F  e.  C  <->  ( F  e.  B  /\  F  =/= 
.0.  /\  ( (coe1 `  F ) `  (
( deg1  `
 R ) `  F ) )  e.  (Unit `  R )
) )
251, 2, 22, 24syl3anbrc 1136 1  |-  ( ( R  e.  DivRing  /\  F  e.  B  /\  F  =/= 
.0.  )  ->  F  e.  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701    =/= wne 2479   -->wf 5288   ` cfv 5292   NN0cn0 10012   Basecbs 13195   0gc0g 13449   Ringcrg 15386  Unitcui 15470   DivRingcdr 15561  Poly1cpl1 16301  coe1cco1 16304   deg1 cdg1 19493  Unic1pcuc1p 19565
This theorem is referenced by:  ig1peu  19610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-addf 8861  ax-mulf 8862
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-sup 7239  df-oi 7270  df-card 7617  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-fz 10830  df-fzo 10918  df-seq 11094  df-hash 11385  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-0g 13453  df-gsum 13454  df-mnd 14416  df-submnd 14465  df-grp 14538  df-minusg 14539  df-mulg 14541  df-subg 14667  df-cntz 14842  df-cmn 15140  df-abl 15141  df-mgp 15375  df-rng 15389  df-cring 15390  df-ur 15391  df-drng 15563  df-psr 16147  df-mpl 16149  df-opsr 16155  df-psr1 16306  df-ply1 16308  df-coe1 16311  df-cnfld 16433  df-mdeg 19494  df-deg1 19495  df-uc1p 19570
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