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Theorem fta1glem1 22858
Description: Lemma for fta1g 22860. (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 1cnd 9642 . 2  |-  ( ph  ->  1  e.  CC )
2 fta1g.1 . . . . . 6  |-  ( ph  ->  R  e. IDomn )
3 isidom 18273 . . . . . . . 8  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
43simprbi 462 . . . . . . 7  |-  ( R  e. IDomn  ->  R  e. Domn )
5 domnnzr 18264 . . . . . . 7  |-  ( R  e. Domn  ->  R  e. NzRing )
64, 5syl 17 . . . . . 6  |-  ( R  e. IDomn  ->  R  e. NzRing )
72, 6syl 17 . . . . 5  |-  ( ph  ->  R  e. NzRing )
8 nzrring 18229 . . . . 5  |-  ( R  e. NzRing  ->  R  e.  Ring )
97, 8syl 17 . . . 4  |-  ( ph  ->  R  e.  Ring )
10 fta1g.2 . . . . 5  |-  ( ph  ->  F  e.  B )
11 fta1g.p . . . . . . . 8  |-  P  =  (Poly1 `  R )
12 fta1g.b . . . . . . . 8  |-  B  =  ( Base `  P
)
13 fta1glem.k . . . . . . . 8  |-  K  =  ( Base `  R
)
14 fta1glem.x . . . . . . . 8  |-  X  =  (var1 `  R )
15 fta1glem.m . . . . . . . 8  |-  .-  =  ( -g `  P )
16 fta1glem.a . . . . . . . 8  |-  A  =  (algSc `  P )
17 fta1glem.g . . . . . . . 8  |-  G  =  ( X  .-  ( A `  T )
)
18 fta1g.o . . . . . . . 8  |-  O  =  (eval1 `  R )
193simplbi 458 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e.  CRing )
202, 19syl 17 . . . . . . . 8  |-  ( ph  ->  R  e.  CRing )
21 fta1glem.5 . . . . . . . . . 10  |-  ( ph  ->  T  e.  ( `' ( O `  F
) " { W } ) )
22 eqid 2402 . . . . . . . . . . . . 13  |-  ( R  ^s  K )  =  ( R  ^s  K )
23 eqid 2402 . . . . . . . . . . . . 13  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
24 fvex 5859 . . . . . . . . . . . . . . 15  |-  ( Base `  R )  e.  _V
2513, 24eqeltri 2486 . . . . . . . . . . . . . 14  |-  K  e. 
_V
2625a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  _V )
2718, 11, 22, 13evl1rhm 18688 . . . . . . . . . . . . . . . 16  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
2820, 27syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
2912, 23rhmf 17695 . . . . . . . . . . . . . . 15  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
3028, 29syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
3130, 10ffvelrnd 6010 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
3222, 13, 23, 2, 26, 31pwselbas 15103 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
) : K --> K )
33 ffn 5714 . . . . . . . . . . . 12  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
3432, 33syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  F
)  Fn  K )
35 fniniseg 5986 . . . . . . . . . . 11  |-  ( ( O `  F )  Fn  K  ->  ( T  e.  ( `' ( O `  F )
" { W }
)  <->  ( T  e.  K  /\  ( ( O `  F ) `
 T )  =  W ) ) )
3634, 35syl 17 . . . . . . . . . 10  |-  ( ph  ->  ( T  e.  ( `' ( O `  F ) " { W } )  <->  ( T  e.  K  /\  (
( O `  F
) `  T )  =  W ) ) )
3721, 36mpbid 210 . . . . . . . . 9  |-  ( ph  ->  ( T  e.  K  /\  ( ( O `  F ) `  T
)  =  W ) )
3837simpld 457 . . . . . . . 8  |-  ( ph  ->  T  e.  K )
39 eqid 2402 . . . . . . . 8  |-  (Monic1p `  R
)  =  (Monic1p `  R
)
40 fta1g.d . . . . . . . 8  |-  D  =  ( deg1  `  R )
41 fta1g.w . . . . . . . 8  |-  W  =  ( 0g `  R
)
4211, 12, 13, 14, 15, 16, 17, 18, 7, 20, 38, 39, 40, 41ply1remlem 22855 . . . . . . 7  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4342simp1d 1009 . . . . . 6  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
44 eqid 2402 . . . . . . 7  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4544, 39mon1puc1p 22843 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
469, 43, 45syl2anc 659 . . . . 5  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
47 eqid 2402 . . . . . 6  |-  (quot1p `  R
)  =  (quot1p `  R
)
4847, 11, 12, 44q1pcl 22848 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
499, 10, 46, 48syl3anc 1230 . . . 4  |-  ( ph  ->  ( F (quot1p `  R
) G )  e.  B )
50 fta1glem.4 . . . . . . . 8  |-  ( ph  ->  ( D `  F
)  =  ( N  +  1 ) )
51 fta1glem.3 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN0 )
52 peano2nn0 10877 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5351, 52syl 17 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
5450, 53eqeltrd 2490 . . . . . . 7  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
55 fta1g.z . . . . . . . . 9  |-  .0.  =  ( 0g `  P )
5640, 11, 55, 12deg1nn0clb 22782 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B )  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 ) )
579, 10, 56syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 ) )
5854, 57mpbird 232 . . . . . 6  |-  ( ph  ->  F  =/=  .0.  )
5937simprd 461 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  F ) `  T
)  =  W )
60 eqid 2402 . . . . . . . . . 10  |-  ( ||r `  P
)  =  ( ||r `  P
)
6111, 12, 13, 14, 15, 16, 17, 18, 7, 20, 38, 10, 41, 60facth1 22857 . . . . . . . . 9  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
6259, 61mpbird 232 . . . . . . . 8  |-  ( ph  ->  G ( ||r `
 P ) F )
63 eqid 2402 . . . . . . . . . 10  |-  ( .r
`  P )  =  ( .r `  P
)
6411, 60, 12, 44, 63, 47dvdsq1p 22853 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( G (
||r `  P ) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
659, 10, 46, 64syl3anc 1230 . . . . . . . 8  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P
) G ) ) )
6662, 65mpbid 210 . . . . . . 7  |-  ( ph  ->  F  =  ( ( F (quot1p `  R ) G ) ( .r `  P ) G ) )
6766eqcomd 2410 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  F )
6811ply1crng 18557 . . . . . . . . 9  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6920, 68syl 17 . . . . . . . 8  |-  ( ph  ->  P  e.  CRing )
70 crngring 17529 . . . . . . . 8  |-  ( P  e.  CRing  ->  P  e.  Ring )
7169, 70syl 17 . . . . . . 7  |-  ( ph  ->  P  e.  Ring )
7211, 12, 39mon1pcl 22837 . . . . . . . 8  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
7343, 72syl 17 . . . . . . 7  |-  ( ph  ->  G  e.  B )
7412, 63, 55ringlz 17555 . . . . . . 7  |-  ( ( P  e.  Ring  /\  G  e.  B )  ->  (  .0.  ( .r `  P
) G )  =  .0.  )
7571, 73, 74syl2anc 659 . . . . . 6  |-  ( ph  ->  (  .0.  ( .r
`  P ) G )  =  .0.  )
7658, 67, 753netr4d 2708 . . . . 5  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =/=  (  .0.  ( .r
`  P ) G ) )
77 oveq1 6285 . . . . . 6  |-  ( ( F (quot1p `  R ) G )  =  .0.  ->  ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =  (  .0.  ( .r `  P ) G ) )
7877necon3i 2643 . . . . 5  |-  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =/=  (  .0.  ( .r `  P
) G )  -> 
( F (quot1p `  R
) G )  =/= 
.0.  )
7976, 78syl 17 . . . 4  |-  ( ph  ->  ( F (quot1p `  R
) G )  =/= 
.0.  )
8040, 11, 55, 12deg1nn0cl 22780 . . . 4  |-  ( ( R  e.  Ring  /\  ( F (quot1p `  R ) G )  e.  B  /\  ( F (quot1p `  R ) G )  =/=  .0.  )  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  NN0 )
819, 49, 79, 80syl3anc 1230 . . 3  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  NN0 )
8281nn0cnd 10895 . 2  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  CC )
8351nn0cnd 10895 . 2  |-  ( ph  ->  N  e.  CC )
8412, 63crngcom 17533 . . . . . . 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 ) ) )
8569, 49, 73, 84syl3anc 1230 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  ( G ( .r
`  P ) ( F (quot1p `  R ) G ) ) )
8666, 85eqtrd 2443 . . . . 5  |-  ( ph  ->  F  =  ( G ( .r `  P
) ( F (quot1p `  R ) G ) ) )
8786fveq2d 5853 . . . 4  |-  ( ph  ->  ( D `  F
)  =  ( D `
 ( G ( .r `  P ) ( F (quot1p `  R
) G ) ) ) )
88 eqid 2402 . . . . 5  |-  (RLReg `  R )  =  (RLReg `  R )
8942simp2d 1010 . . . . . . 7  |-  ( ph  ->  ( D `  G
)  =  1 )
90 1nn0 10852 . . . . . . 7  |-  1  e.  NN0
9189, 90syl6eqel 2498 . . . . . 6  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
9240, 11, 55, 12deg1nn0clb 22782 . . . . . . 7  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  .0.  <->  ( D `  G )  e.  NN0 ) )
939, 73, 92syl2anc 659 . . . . . 6  |-  ( ph  ->  ( G  =/=  .0.  <->  ( D `  G )  e.  NN0 ) )
9491, 93mpbird 232 . . . . 5  |-  ( ph  ->  G  =/=  .0.  )
95 eqid 2402 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
9688, 95unitrrg 18262 . . . . . . 7  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
979, 96syl 17 . . . . . 6  |-  ( ph  ->  (Unit `  R )  C_  (RLReg `  R )
)
9840, 95, 44uc1pldg 22841 . . . . . . 7  |-  ( G  e.  (Unic1p `  R )  -> 
( (coe1 `  G ) `  ( D `  G ) )  e.  (Unit `  R ) )
9946, 98syl 17 . . . . . 6  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (Unit `  R ) )
10097, 99sseldd 3443 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (RLReg `  R ) )
10140, 11, 88, 12, 63, 55, 9, 73, 94, 100, 49, 79deg1mul2 22807 . . . 4  |-  ( ph  ->  ( D `  ( G ( .r `  P ) ( F (quot1p `  R ) G ) ) )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
10287, 50, 1013eqtr3d 2451 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
103 ax-1cn 9580 . . . 4  |-  1  e.  CC
104 addcom 9800 . . . 4  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  +  1 )  =  ( 1  +  N ) )
10583, 103, 104sylancl 660 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( 1  +  N ) )
10689oveq1d 6293 . . 3  |-  ( ph  ->  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
107102, 105, 1063eqtr3rd 2452 . 2  |-  ( ph  ->  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  N
) )
1081, 82, 83, 107addcanad 9819 1  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   _Vcvv 3059    C_ wss 3414   {csn 3972   class class class wbr 4395   `'ccnv 4822   "cima 4826    Fn wfn 5564   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   1c1 9523    + caddc 9525   NN0cn0 10836   Basecbs 14841   .rcmulr 14910   0gc0g 15054    ^s cpws 15061   -gcsg 16379   Ringcrg 17518   CRingccrg 17519   ||rcdsr 17607  Unitcui 17608   RingHom crh 17681  NzRingcnzr 18225  RLRegcrlreg 18247  Domncdomn 18248  IDomncidom 18249  algSccascl 18280  var1cv1 18535  Poly1cpl1 18536  coe1cco1 18537  eval1ce1 18671   deg1 cdg1 22744  Monic1pcmn1 22818  Unic1pcuc1p 22819  quot1pcq1p 22820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-ofr 6522  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-tpos 6958  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-fz 11727  df-fzo 11855  df-seq 12152  df-hash 12453  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-0g 15056  df-gsum 15057  df-prds 15062  df-pws 15064  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-mhm 16290  df-submnd 16291  df-grp 16381  df-minusg 16382  df-sbg 16383  df-mulg 16384  df-subg 16522  df-ghm 16589  df-cntz 16679  df-cmn 17124  df-abl 17125  df-mgp 17462  df-ur 17474  df-srg 17478  df-ring 17520  df-cring 17521  df-oppr 17592  df-dvdsr 17610  df-unit 17611  df-invr 17641  df-rnghom 17684  df-subrg 17747  df-lmod 17834  df-lss 17899  df-lsp 17938  df-nzr 18226  df-rlreg 18251  df-domn 18252  df-idom 18253  df-assa 18281  df-asp 18282  df-ascl 18283  df-psr 18325  df-mvr 18326  df-mpl 18327  df-opsr 18329  df-evls 18491  df-evl 18492  df-psr1 18539  df-vr1 18540  df-ply1 18541  df-coe1 18542  df-evl1 18673  df-cnfld 18741  df-mdeg 22745  df-deg1 22746  df-mon1 22823  df-uc1p 22824  df-q1p 22825  df-r1p 22826
This theorem is referenced by:  fta1glem2  22859
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