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Theorem fta1glem1 20088
Description: Lemma for fta1g 20090. (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 9048 . . 3  |-  1  e.  CC
21a1i 11 . 2  |-  ( ph  ->  1  e.  CC )
3 fta1g.1 . . . . . 6  |-  ( ph  ->  R  e. IDomn )
4 isidom 16364 . . . . . . . 8  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
54simprbi 451 . . . . . . 7  |-  ( R  e. IDomn  ->  R  e. Domn )
6 domnnzr 16355 . . . . . . 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 16332 . . . . 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 2436 . . . . . . . . . . . . 13  |-  ( R  ^s  K )  =  ( R  ^s  K )
24 eqid 2436 . . . . . . . . . . . . 13  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
25 fvex 5742 . . . . . . . . . . . . . . 15  |-  ( Base `  R )  e.  _V
2614, 25eqeltri 2506 . . . . . . . . . . . . . 14  |-  K  e. 
_V
2726a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  _V )
2819, 12, 23, 14evl1rhm 19949 . . . . . . . . . . . . . . . 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 15827 . . . . . . . . . . . . . . 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 5871 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
3323, 14, 24, 3, 27, 32pwselbas 13711 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
) : K --> K )
34 ffn 5591 . . . . . . . . . . . 12  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
3533, 34syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  F
)  Fn  K )
36 fniniseg 5851 . . . . . . . . . . 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 2436 . . . . . . . 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 20085 . . . . . . 7  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4443simp1d 969 . . . . . 6  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
45 eqid 2436 . . . . . . 7  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4645, 40mon1puc1p 20073 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
4710, 44, 46syl2anc 643 . . . . 5  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
48 eqid 2436 . . . . . 6  |-  (quot1p `  R
)  =  (quot1p `  R
)
4948, 12, 13, 45q1pcl 20078 . . . . 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 10260 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5452, 53syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
5551, 54eqeltrd 2510 . . . . . . 7  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
56 fta1g.z . . . . . . . . 9  |-  .0.  =  ( 0g `  P )
5741, 12, 56, 13deg1nn0clb 20013 . . . . . . . 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 2436 . . . . . . . . . 10  |-  ( ||r `  P
)  =  ( ||r `  P
)
6212, 13, 14, 15, 16, 17, 18, 19, 8, 21, 39, 11, 42, 61facth1 20087 . . . . . . . . 9  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
6360, 62mpbird 224 . . . . . . . 8  |-  ( ph  ->  G ( ||r `
 P ) F )
64 eqid 2436 . . . . . . . . . 10  |-  ( .r
`  P )  =  ( .r `  P
)
6512, 61, 13, 45, 64, 48dvdsq1p 20083 . . . . . . . . 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 2441 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  F )
6912ply1crng 16596 . . . . . . . . 9  |-  ( R  e.  CRing  ->  P  e.  CRing
)
7021, 69syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  CRing )
71 crngrng 15674 . . . . . . . 8  |-  ( P  e.  CRing  ->  P  e.  Ring )
7270, 71syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  Ring )
7312, 13, 40mon1pcl 20067 . . . . . . . 8  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
7444, 73syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  B )
7513, 64, 56rnglz 15700 . . . . . . 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 2628 . . . . 5  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =/=  (  .0.  ( .r
`  P ) G ) )
78 oveq1 6088 . . . . . 6  |-  ( ( F (quot1p `  R ) G )  =  .0.  ->  ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =  (  .0.  ( .r `  P ) G ) )
7978necon3i 2643 . . . . 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 20011 . . . 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 10276 . 2  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  CC )
8452nn0cnd 10276 . 2  |-  ( ph  ->  N  e.  CC )
8513, 64crngcom 15678 . . . . . . 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 2468 . . . . 5  |-  ( ph  ->  F  =  ( G ( .r `  P
) ( F (quot1p `  R ) G ) ) )
8887fveq2d 5732 . . . 4  |-  ( ph  ->  ( D `  F
)  =  ( D `
 ( G ( .r `  P ) ( F (quot1p `  R
) G ) ) ) )
89 eqid 2436 . . . . 5  |-  (RLReg `  R )  =  (RLReg `  R )
9043simp2d 970 . . . . . . 7  |-  ( ph  ->  ( D `  G
)  =  1 )
91 1nn0 10237 . . . . . . 7  |-  1  e.  NN0
9290, 91syl6eqel 2524 . . . . . 6  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
9341, 12, 56, 13deg1nn0clb 20013 . . . . . . 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 2436 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
9789, 96unitrrg 16353 . . . . . . 7  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
9810, 97syl 16 . . . . . 6  |-  ( ph  ->  (Unit `  R )  C_  (RLReg `  R )
)
9941, 96, 45uc1pldg 20071 . . . . . . 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 3349 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (RLReg `  R ) )
10241, 12, 89, 13, 64, 56, 10, 74, 95, 101, 50, 80deg1mul2 20037 . . . 4  |-  ( ph  ->  ( D `  ( G ( .r `  P ) ( F (quot1p `  R ) G ) ) )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
10388, 51, 1023eqtr3d 2476 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
104 addcom 9252 . . . 4  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  +  1 )  =  ( 1  +  N ) )
10584, 1, 104sylancl 644 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( 1  +  N ) )
10690oveq1d 6096 . . 3  |-  ( ph  ->  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
107103, 105, 1063eqtr3rd 2477 . 2  |-  ( ph  ->  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  N
) )
1082, 83, 84, 107addcanad 9271 1  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956    C_ wss 3320   {csn 3814   class class class wbr 4212   `'ccnv 4877   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   CCcc 8988   1c1 8991    + caddc 8993   NN0cn0 10221   Basecbs 13469   .rcmulr 13530    ^s cpws 13670   0gc0g 13723   -gcsg 14688   Ringcrg 15660   CRingccrg 15661   ||rcdsr 15743  Unitcui 15744   RingHom crh 15817  NzRingcnzr 16328  RLRegcrlreg 16339  Domncdomn 16340  IDomncidom 16341  algSccascl 16371  var1cv1 16570  Poly1cpl1 16571  eval1ce1 16573  coe1cco1 16574   deg1 cdg1 19977  Monic1pcmn1 20048  Unic1pcuc1p 20049  quot1pcq1p 20050
This theorem is referenced by:  fta1glem2  20089
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-ofr 6306  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-pm 7021  df-ixp 7064  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-fz 11044  df-fzo 11136  df-seq 11324  df-hash 11619  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-sca 13545  df-vsca 13546  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-hom 13553  df-cco 13554  df-prds 13671  df-pws 13673  df-0g 13727  df-gsum 13728  df-mre 13811  df-mrc 13812  df-acs 13814  df-mnd 14690  df-mhm 14738  df-submnd 14739  df-grp 14812  df-minusg 14813  df-sbg 14814  df-mulg 14815  df-subg 14941  df-ghm 15004  df-cntz 15116  df-cmn 15414  df-abl 15415  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-rnghom 15819  df-subrg 15866  df-lmod 15952  df-lss 16009  df-lsp 16048  df-nzr 16329  df-rlreg 16343  df-domn 16344  df-idom 16345  df-assa 16372  df-asp 16373  df-ascl 16374  df-psr 16417  df-mvr 16418  df-mpl 16419  df-evls 16420  df-evl 16421  df-opsr 16425  df-psr1 16576  df-vr1 16577  df-ply1 16578  df-evl1 16580  df-coe1 16581  df-cnfld 16704  df-mdeg 19978  df-deg1 19979  df-mon1 20053  df-uc1p 20054  df-q1p 20055  df-r1p 20056
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