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Theorem fta1glem1 20041
Description: Lemma for fta1g 20043. (Contributed by Mario Carneiro, 7-Jun-2016.)
Hypotheses
Ref Expression
fta1g.p  |-  P  =  (Poly1 `  R )
fta1g.b  |-  B  =  ( Base `  P
)
fta1g.d  |-  D  =  ( deg1  `  R )
fta1g.o  |-  O  =  (eval1 `  R )
fta1g.w  |-  W  =  ( 0g `  R
)
fta1g.z  |-  .0.  =  ( 0g `  P )
fta1g.1  |-  ( ph  ->  R  e. IDomn )
fta1g.2  |-  ( ph  ->  F  e.  B )
fta1glem.k  |-  K  =  ( Base `  R
)
fta1glem.x  |-  X  =  (var1 `  R )
fta1glem.m  |-  .-  =  ( -g `  P )
fta1glem.a  |-  A  =  (algSc `  P )
fta1glem.g  |-  G  =  ( X  .-  ( A `  T )
)
fta1glem.3  |-  ( ph  ->  N  e.  NN0 )
fta1glem.4  |-  ( ph  ->  ( D `  F
)  =  ( N  +  1 ) )
fta1glem.5  |-  ( ph  ->  T  e.  ( `' ( O `  F
) " { W } ) )
Assertion
Ref Expression
fta1glem1  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )

Proof of Theorem fta1glem1
StepHypRef Expression
1 ax-1cn 9004 . . 3  |-  1  e.  CC
21a1i 11 . 2  |-  ( ph  ->  1  e.  CC )
3 fta1g.1 . . . . . 6  |-  ( ph  ->  R  e. IDomn )
4 isidom 16319 . . . . . . . 8  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
54simprbi 451 . . . . . . 7  |-  ( R  e. IDomn  ->  R  e. Domn )
6 domnnzr 16310 . . . . . . 7  |-  ( R  e. Domn  ->  R  e. NzRing )
75, 6syl 16 . . . . . 6  |-  ( R  e. IDomn  ->  R  e. NzRing )
83, 7syl 16 . . . . 5  |-  ( ph  ->  R  e. NzRing )
9 nzrrng 16287 . . . . 5  |-  ( R  e. NzRing  ->  R  e.  Ring )
108, 9syl 16 . . . 4  |-  ( ph  ->  R  e.  Ring )
11 fta1g.2 . . . . 5  |-  ( ph  ->  F  e.  B )
12 fta1g.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
13 fta1g.b . . . . . . . 8  |-  B  =  ( Base `  P
)
14 fta1glem.k . . . . . . . 8  |-  K  =  ( Base `  R
)
15 fta1glem.x . . . . . . . 8  |-  X  =  (var1 `  R )
16 fta1glem.m . . . . . . . 8  |-  .-  =  ( -g `  P )
17 fta1glem.a . . . . . . . 8  |-  A  =  (algSc `  P )
18 fta1glem.g . . . . . . . 8  |-  G  =  ( X  .-  ( A `  T )
)
19 fta1g.o . . . . . . . 8  |-  O  =  (eval1 `  R )
204simplbi 447 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e.  CRing )
213, 20syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  CRing )
22 fta1glem.5 . . . . . . . . . 10  |-  ( ph  ->  T  e.  ( `' ( O `  F
) " { W } ) )
23 eqid 2404 . . . . . . . . . . . . 13  |-  ( R  ^s  K )  =  ( R  ^s  K )
24 eqid 2404 . . . . . . . . . . . . 13  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
25 fvex 5701 . . . . . . . . . . . . . . 15  |-  ( Base `  R )  e.  _V
2614, 25eqeltri 2474 . . . . . . . . . . . . . 14  |-  K  e. 
_V
2726a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  _V )
2819, 12, 23, 14evl1rhm 19902 . . . . . . . . . . . . . . . 16  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
2921, 28syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
3013, 24rhmf 15782 . . . . . . . . . . . . . . 15  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
3129, 30syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
3231, 11ffvelrnd 5830 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
3323, 14, 24, 3, 27, 32pwselbas 13666 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
) : K --> K )
34 ffn 5550 . . . . . . . . . . . 12  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
3533, 34syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  F
)  Fn  K )
36 fniniseg 5810 . . . . . . . . . . 11  |-  ( ( O `  F )  Fn  K  ->  ( T  e.  ( `' ( O `  F )
" { W }
)  <->  ( T  e.  K  /\  ( ( O `  F ) `
 T )  =  W ) ) )
3735, 36syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( T  e.  ( `' ( O `  F ) " { W } )  <->  ( T  e.  K  /\  (
( O `  F
) `  T )  =  W ) ) )
3822, 37mpbid 202 . . . . . . . . 9  |-  ( ph  ->  ( T  e.  K  /\  ( ( O `  F ) `  T
)  =  W ) )
3938simpld 446 . . . . . . . 8  |-  ( ph  ->  T  e.  K )
40 eqid 2404 . . . . . . . 8  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
41 fta1g.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
42 fta1g.w . . . . . . . 8  |-  W  =  ( 0g `  R
)
4312, 13, 14, 15, 16, 17, 18, 19, 8, 21, 39, 40, 41, 42ply1remlem 20038 . . . . . . 7  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4443simp1d 969 . . . . . 6  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
45 eqid 2404 . . . . . . 7  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4645, 40mon1puc1p 20026 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
4710, 44, 46syl2anc 643 . . . . 5  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
48 eqid 2404 . . . . . 6  |-  (quot1p `  R
)  =  (quot1p `  R
)
4948, 12, 13, 45q1pcl 20031 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
5010, 11, 47, 49syl3anc 1184 . . . 4  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
51 fta1glem.4 . . . . . . . 8  |-  ( ph  ->  ( D `  F
)  =  ( N  +  1 ) )
52 fta1glem.3 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
53 peano2nn0 10216 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5452, 53syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
5551, 54eqeltrd 2478 . . . . . . 7  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
56 fta1g.z . . . . . . . . 9  |-  .0.  =  ( 0g `  P )
5741, 12, 56, 13deg1nn0clb 19966 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B )  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 ) )
5810, 11, 57syl2anc 643 . . . . . . 7  |-  ( ph  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 ) )
5955, 58mpbird 224 . . . . . 6  |-  ( ph  ->  F  =/=  .0.  )
6038simprd 450 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  F ) `  T
)  =  W )
61 eqid 2404 . . . . . . . . . 10  |-  ( ||r `  P
)  =  ( ||r `  P
)
6212, 13, 14, 15, 16, 17, 18, 19, 8, 21, 39, 11, 42, 61facth1 20040 . . . . . . . . 9  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
6360, 62mpbird 224 . . . . . . . 8  |-  ( ph  ->  G ( ||r `
 P ) F )
64 eqid 2404 . . . . . . . . . 10  |-  ( .r
`  P )  =  ( .r `  P
)
6512, 61, 13, 45, 64, 48dvdsq1p 20036 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( G (
||r `  P ) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
6610, 11, 47, 65syl3anc 1184 . . . . . . . 8  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
6763, 66mpbid 202 . . . . . . 7  |-  ( ph  ->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )
6867eqcomd 2409 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  F )
6912ply1crng 16551 . . . . . . . . 9  |-  ( R  e.  CRing  ->  P  e.  CRing
)
7021, 69syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  CRing )
71 crngrng 15629 . . . . . . . 8  |-  ( P  e.  CRing  ->  P  e.  Ring )
7270, 71syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  Ring )
7312, 13, 40mon1pcl 20020 . . . . . . . 8  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
7444, 73syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  B )
7513, 64, 56rnglz 15655 . . . . . . 7  |-  ( ( P  e.  Ring  /\  G  e.  B )  ->  (  .0.  ( .r `  P
) G )  =  .0.  )
7672, 74, 75syl2anc 643 . . . . . 6  |-  ( ph  ->  (  .0.  ( .r
`  P ) G )  =  .0.  )
7759, 68, 763netr4d 2594 . . . . 5  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =/=  (  .0.  ( .r
`  P ) G ) )
78 oveq1 6047 . . . . . 6  |-  ( ( F (quot1p `  R ) G )  =  .0.  ->  ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =  (  .0.  ( .r `  P ) G ) )
7978necon3i 2606 . . . . 5  |-  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =/=  (  .0.  ( .r `  P
) G )  -> 
( F (quot1p `  R
) G )  =/= 
.0.  )
8077, 79syl 16 . . . 4  |-  ( ph  ->  ( F (quot1p `  R
) G )  =/= 
.0.  )
8141, 12, 56, 13deg1nn0cl 19964 . . . 4  |-  ( ( R  e.  Ring  /\  ( F (quot1p `  R ) G )  e.  B  /\  ( F (quot1p `  R ) G )  =/=  .0.  )  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  NN0 )
8210, 50, 80, 81syl3anc 1184 . . 3  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  NN0 )
8382nn0cnd 10232 . 2  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  CC )
8452nn0cnd 10232 . 2  |-  ( ph  ->  N  e.  CC )
8513, 64crngcom 15633 . . . . . . 7  |-  ( ( P  e.  CRing  /\  ( F (quot1p `  R ) G )  e.  B  /\  G  e.  B )  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  ( G ( .r
`  P ) ( F (quot1p `  R ) G ) ) )
8670, 50, 74, 85syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  ( G ( .r
`  P ) ( F (quot1p `  R ) G ) ) )
8767, 86eqtrd 2436 . . . . 5  |-  ( ph  ->  F  =  ( G ( .r `  P
) ( F (quot1p `  R ) G ) ) )
8887fveq2d 5691 . . . 4  |-  ( ph  ->  ( D `  F
)  =  ( D `
 ( G ( .r `  P ) ( F (quot1p `  R
) G ) ) ) )
89 eqid 2404 . . . . 5  |-  (RLReg `  R )  =  (RLReg `  R )
9043simp2d 970 . . . . . . 7  |-  ( ph  ->  ( D `  G
)  =  1 )
91 1nn0 10193 . . . . . . 7  |-  1  e.  NN0
9290, 91syl6eqel 2492 . . . . . 6  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
9341, 12, 56, 13deg1nn0clb 19966 . . . . . . 7  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  .0.  <->  ( D `  G )  e.  NN0 ) )
9410, 74, 93syl2anc 643 . . . . . 6  |-  ( ph  ->  ( G  =/=  .0.  <->  ( D `  G )  e.  NN0 ) )
9592, 94mpbird 224 . . . . 5  |-  ( ph  ->  G  =/=  .0.  )
96 eqid 2404 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
9789, 96unitrrg 16308 . . . . . . 7  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
9810, 97syl 16 . . . . . 6  |-  ( ph  ->  (Unit `  R )  C_  (RLReg `  R )
)
9941, 96, 45uc1pldg 20024 . . . . . . 7  |-  ( G  e.  (Unic1p `  R )  -> 
( (coe1 `  G ) `  ( D `  G ) )  e.  (Unit `  R ) )
10047, 99syl 16 . . . . . 6  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (Unit `  R ) )
10198, 100sseldd 3309 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (RLReg `  R ) )
10241, 12, 89, 13, 64, 56, 10, 74, 95, 101, 50, 80deg1mul2 19990 . . . 4  |-  ( ph  ->  ( D `  ( G ( .r `  P ) ( F (quot1p `  R ) G ) ) )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
10388, 51, 1023eqtr3d 2444 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
104 addcom 9208 . . . 4  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  +  1 )  =  ( 1  +  N ) )
10584, 1, 104sylancl 644 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( 1  +  N ) )
10690oveq1d 6055 . . 3  |-  ( ph  ->  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
107103, 105, 1063eqtr3rd 2445 . 2  |-  ( ph  ->  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  N
) )
1082, 83, 84, 107addcanad 9227 1  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2567   _Vcvv 2916    C_ wss 3280   {csn 3774   class class class wbr 4172   `'ccnv 4836   "cima 4840    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   1c1 8947    + caddc 8949   NN0cn0 10177   Basecbs 13424   .rcmulr 13485    ^s cpws 13625   0gc0g 13678   -gcsg 14643   Ringcrg 15615   CRingccrg 15616   ||rcdsr 15698  Unitcui 15699   RingHom crh 15772  NzRingcnzr 16283  RLRegcrlreg 16294  Domncdomn 16295  IDomncidom 16296  algSccascl 16326  var1cv1 16525  Poly1cpl1 16526  eval1ce1 16528  coe1cco1 16529   deg1 cdg1 19930  Monic1pcmn1 20001  Unic1pcuc1p 20002  quot1pcq1p 20003
This theorem is referenced by:  fta1glem2  20042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-starv 13499  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-ds 13506  df-unif 13507  df-hom 13508  df-cco 13509  df-prds 13626  df-pws 13628  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-cring 15619  df-ur 15620  df-oppr 15683  df-dvdsr 15701  df-unit 15702  df-invr 15732  df-rnghom 15774  df-subrg 15821  df-lmod 15907  df-lss 15964  df-lsp 16003  df-nzr 16284  df-rlreg 16298  df-domn 16299  df-idom 16300  df-assa 16327  df-asp 16328  df-ascl 16329  df-psr 16372  df-mvr 16373  df-mpl 16374  df-evls 16375  df-evl 16376  df-opsr 16380  df-psr1 16531  df-vr1 16532  df-ply1 16533  df-evl1 16535  df-coe1 16536  df-cnfld 16659  df-mdeg 19931  df-deg1 19932  df-mon1 20006  df-uc1p 20007  df-q1p 20008  df-r1p 20009
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