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Theorem fta1glem1 21642
Description: Lemma for fta1g 21644. (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 9407 . 2  |-  ( ph  ->  1  e.  CC )
2 fta1g.1 . . . . . 6  |-  ( ph  ->  R  e. IDomn )
3 isidom 17381 . . . . . . . 8  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
43simprbi 464 . . . . . . 7  |-  ( R  e. IDomn  ->  R  e. Domn )
5 domnnzr 17372 . . . . . . 7  |-  ( R  e. Domn  ->  R  e. NzRing )
64, 5syl 16 . . . . . 6  |-  ( R  e. IDomn  ->  R  e. NzRing )
72, 6syl 16 . . . . 5  |-  ( ph  ->  R  e. NzRing )
8 nzrrng 17348 . . . . 5  |-  ( R  e. NzRing  ->  R  e.  Ring )
97, 8syl 16 . . . 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 460 . . . . . . . . 9  |-  ( R  e. IDomn  ->  R  e.  CRing )
202, 19syl 16 . . . . . . . 8  |-  ( ph  ->  R  e.  CRing )
21 fta1glem.5 . . . . . . . . . 10  |-  ( ph  ->  T  e.  ( `' ( O `  F
) " { W } ) )
22 eqid 2443 . . . . . . . . . . . . 13  |-  ( R  ^s  K )  =  ( R  ^s  K )
23 eqid 2443 . . . . . . . . . . . . 13  |-  ( Base `  ( R  ^s  K ) )  =  ( Base `  ( R  ^s  K ) )
24 fvex 5706 . . . . . . . . . . . . . . 15  |-  ( Base `  R )  e.  _V
2513, 24eqeltri 2513 . . . . . . . . . . . . . 14  |-  K  e. 
_V
2625a1i 11 . . . . . . . . . . . . 13  |-  ( ph  ->  K  e.  _V )
2718, 11, 22, 13evl1rhm 17771 . . . . . . . . . . . . . . . 16  |-  ( R  e.  CRing  ->  O  e.  ( P RingHom  ( R  ^s  K
) ) )
2820, 27syl 16 . . . . . . . . . . . . . . 15  |-  ( ph  ->  O  e.  ( P RingHom 
( R  ^s  K ) ) )
2912, 23rhmf 16821 . . . . . . . . . . . . . . 15  |-  ( O  e.  ( P RingHom  ( R  ^s  K ) )  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
3028, 29syl 16 . . . . . . . . . . . . . 14  |-  ( ph  ->  O : B --> ( Base `  ( R  ^s  K ) ) )
3130, 10ffvelrnd 5849 . . . . . . . . . . . . 13  |-  ( ph  ->  ( O `  F
)  e.  ( Base `  ( R  ^s  K ) ) )
3222, 13, 23, 2, 26, 31pwselbas 14432 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  F
) : K --> K )
33 ffn 5564 . . . . . . . . . . . 12  |-  ( ( O `  F ) : K --> K  -> 
( O `  F
)  Fn  K )
3432, 33syl 16 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  F
)  Fn  K )
35 fniniseg 5829 . . . . . . . . . . 11  |-  ( ( O `  F )  Fn  K  ->  ( T  e.  ( `' ( O `  F )
" { W }
)  <->  ( T  e.  K  /\  ( ( O `  F ) `
 T )  =  W ) ) )
3634, 35syl 16 . . . . . . . . . 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 459 . . . . . . . 8  |-  ( ph  ->  T  e.  K )
39 eqid 2443 . . . . . . . 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 21639 . . . . . . 7  |-  ( ph  ->  ( G  e.  (Monic1p `  R )  /\  ( D `  G )  =  1  /\  ( `' ( O `  G ) " { W } )  =  { T } ) )
4342simp1d 1000 . . . . . 6  |-  ( ph  ->  G  e.  (Monic1p `  R
) )
44 eqid 2443 . . . . . . 7  |-  (Unic1p `  R
)  =  (Unic1p `  R
)
4544, 39mon1puc1p 21627 . . . . . 6  |-  ( ( R  e.  Ring  /\  G  e.  (Monic1p `  R ) )  ->  G  e.  (Unic1p `  R ) )
469, 43, 45syl2anc 661 . . . . 5  |-  ( ph  ->  G  e.  (Unic1p `  R
) )
47 eqid 2443 . . . . . 6  |-  (quot1p `  R
)  =  (quot1p `  R
)
4847, 11, 12, 44q1pcl 21632 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  (Unic1p `  R ) )  ->  ( F (quot1p `  R ) G )  e.  B )
499, 10, 46, 48syl3anc 1218 . . . 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 10625 . . . . . . . . 9  |-  ( N  e.  NN0  ->  ( N  +  1 )  e. 
NN0 )
5351, 52syl 16 . . . . . . . 8  |-  ( ph  ->  ( N  +  1 )  e.  NN0 )
5450, 53eqeltrd 2517 . . . . . . 7  |-  ( ph  ->  ( D `  F
)  e.  NN0 )
55 fta1g.z . . . . . . . . 9  |-  .0.  =  ( 0g `  P )
5640, 11, 55, 12deg1nn0clb 21566 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B )  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 ) )
579, 10, 56syl2anc 661 . . . . . . 7  |-  ( ph  ->  ( F  =/=  .0.  <->  ( D `  F )  e.  NN0 ) )
5854, 57mpbird 232 . . . . . 6  |-  ( ph  ->  F  =/=  .0.  )
5937simprd 463 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  F ) `  T
)  =  W )
60 eqid 2443 . . . . . . . . . 10  |-  ( ||r `  P
)  =  ( ||r `  P
)
6111, 12, 13, 14, 15, 16, 17, 18, 7, 20, 38, 10, 41, 60facth1 21641 . . . . . . . . 9  |-  ( ph  ->  ( G ( ||r `  P
) F  <->  ( ( O `  F ) `  T )  =  W ) )
6259, 61mpbird 232 . . . . . . . 8  |-  ( ph  ->  G ( ||r `
 P ) F )
63 eqid 2443 . . . . . . . . . 10  |-  ( .r
`  P )  =  ( .r `  P
)
6411, 60, 12, 44, 63, 47dvdsq1p 21637 . . . . . . . . 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 1218 . . . . . . . 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 2448 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  F )
6811ply1crng 17659 . . . . . . . . 9  |-  ( R  e.  CRing  ->  P  e.  CRing
)
6920, 68syl 16 . . . . . . . 8  |-  ( ph  ->  P  e.  CRing )
70 crngrng 16660 . . . . . . . 8  |-  ( P  e.  CRing  ->  P  e.  Ring )
7169, 70syl 16 . . . . . . 7  |-  ( ph  ->  P  e.  Ring )
7211, 12, 39mon1pcl 21621 . . . . . . . 8  |-  ( G  e.  (Monic1p `  R )  ->  G  e.  B )
7343, 72syl 16 . . . . . . 7  |-  ( ph  ->  G  e.  B )
7412, 63, 55rnglz 16686 . . . . . . 7  |-  ( ( P  e.  Ring  /\  G  e.  B )  ->  (  .0.  ( .r `  P
) G )  =  .0.  )
7571, 73, 74syl2anc 661 . . . . . 6  |-  ( ph  ->  (  .0.  ( .r
`  P ) G )  =  .0.  )
7658, 67, 753netr4d 2640 . . . . 5  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =/=  (  .0.  ( .r
`  P ) G ) )
77 oveq1 6103 . . . . . 6  |-  ( ( F (quot1p `  R ) G )  =  .0.  ->  ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =  (  .0.  ( .r `  P ) G ) )
7877necon3i 2655 . . . . 5  |-  ( ( ( F (quot1p `  R
) G ) ( .r `  P ) G )  =/=  (  .0.  ( .r `  P
) G )  -> 
( F (quot1p `  R
) G )  =/= 
.0.  )
7976, 78syl 16 . . . 4  |-  ( ph  ->  ( F (quot1p `  R
) G )  =/= 
.0.  )
8040, 11, 55, 12deg1nn0cl 21564 . . . 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 1218 . . 3  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  NN0 )
8281nn0cnd 10643 . 2  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  e.  CC )
8351nn0cnd 10643 . 2  |-  ( ph  ->  N  e.  CC )
8412, 63crngcom 16664 . . . . . . 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 1218 . . . . . 6  |-  ( ph  ->  ( ( F (quot1p `  R ) G ) ( .r `  P
) G )  =  ( G ( .r
`  P ) ( F (quot1p `  R ) G ) ) )
8666, 85eqtrd 2475 . . . . 5  |-  ( ph  ->  F  =  ( G ( .r `  P
) ( F (quot1p `  R ) G ) ) )
8786fveq2d 5700 . . . 4  |-  ( ph  ->  ( D `  F
)  =  ( D `
 ( G ( .r `  P ) ( F (quot1p `  R
) G ) ) ) )
88 eqid 2443 . . . . 5  |-  (RLReg `  R )  =  (RLReg `  R )
8942simp2d 1001 . . . . . . 7  |-  ( ph  ->  ( D `  G
)  =  1 )
90 1nn0 10600 . . . . . . 7  |-  1  e.  NN0
9189, 90syl6eqel 2531 . . . . . 6  |-  ( ph  ->  ( D `  G
)  e.  NN0 )
9240, 11, 55, 12deg1nn0clb 21566 . . . . . . 7  |-  ( ( R  e.  Ring  /\  G  e.  B )  ->  ( G  =/=  .0.  <->  ( D `  G )  e.  NN0 ) )
939, 73, 92syl2anc 661 . . . . . 6  |-  ( ph  ->  ( G  =/=  .0.  <->  ( D `  G )  e.  NN0 ) )
9491, 93mpbird 232 . . . . 5  |-  ( ph  ->  G  =/=  .0.  )
95 eqid 2443 . . . . . . . 8  |-  (Unit `  R )  =  (Unit `  R )
9688, 95unitrrg 17370 . . . . . . 7  |-  ( R  e.  Ring  ->  (Unit `  R )  C_  (RLReg `  R ) )
979, 96syl 16 . . . . . 6  |-  ( ph  ->  (Unit `  R )  C_  (RLReg `  R )
)
9840, 95, 44uc1pldg 21625 . . . . . . 7  |-  ( G  e.  (Unic1p `  R )  -> 
( (coe1 `  G ) `  ( D `  G ) )  e.  (Unit `  R ) )
9946, 98syl 16 . . . . . 6  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (Unit `  R ) )
10097, 99sseldd 3362 . . . . 5  |-  ( ph  ->  ( (coe1 `  G ) `  ( D `  G ) )  e.  (RLReg `  R ) )
10140, 11, 88, 12, 63, 55, 9, 73, 94, 100, 49, 79deg1mul2 21591 . . . 4  |-  ( ph  ->  ( D `  ( G ( .r `  P ) ( F (quot1p `  R ) G ) ) )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
10287, 50, 1013eqtr3d 2483 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
103 ax-1cn 9345 . . . 4  |-  1  e.  CC
104 addcom 9560 . . . 4  |-  ( ( N  e.  CC  /\  1  e.  CC )  ->  ( N  +  1 )  =  ( 1  +  N ) )
10583, 103, 104sylancl 662 . . 3  |-  ( ph  ->  ( N  +  1 )  =  ( 1  +  N ) )
10689oveq1d 6111 . . 3  |-  ( ph  ->  ( ( D `  G )  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) ) )
107102, 105, 1063eqtr3rd 2484 . 2  |-  ( ph  ->  ( 1  +  ( D `  ( F (quot1p `  R ) G ) ) )  =  ( 1  +  N
) )
1081, 82, 83, 107addcanad 9579 1  |-  ( ph  ->  ( D `  ( F (quot1p `  R ) G ) )  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611   _Vcvv 2977    C_ wss 3333   {csn 3882   class class class wbr 4297   `'ccnv 4844   "cima 4848    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   1c1 9288    + caddc 9290   NN0cn0 10584   Basecbs 14179   .rcmulr 14244   0gc0g 14383    ^s cpws 14390   -gcsg 15418   Ringcrg 16650   CRingccrg 16651   ||rcdsr 16735  Unitcui 16736   RingHom crh 16809  NzRingcnzr 17344  RLRegcrlreg 17355  Domncdomn 17356  IDomncidom 17357  algSccascl 17388  var1cv1 17637  Poly1cpl1 17638  coe1cco1 17639  eval1ce1 17754   deg1 cdg1 21528  Monic1pcmn1 21602  Unic1pcuc1p 21603  quot1pcq1p 21604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-inf2 7852  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364  ax-pre-sup 9365  ax-addf 9366  ax-mulf 9367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-iin 4179  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-se 4685  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-isom 5432  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-ofr 6326  df-om 6482  df-1st 6582  df-2nd 6583  df-supp 6696  df-tpos 6750  df-recs 6837  df-rdg 6871  df-1o 6925  df-2o 6926  df-oadd 6929  df-er 7106  df-map 7221  df-pm 7222  df-ixp 7269  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-fsupp 7626  df-sup 7696  df-oi 7729  df-card 8114  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-2 10385  df-3 10386  df-4 10387  df-5 10388  df-6 10389  df-7 10390  df-8 10391  df-9 10392  df-10 10393  df-n0 10585  df-z 10652  df-dec 10761  df-uz 10867  df-fz 11443  df-fzo 11554  df-seq 11812  df-hash 12109  df-struct 14181  df-ndx 14182  df-slot 14183  df-base 14184  df-sets 14185  df-ress 14186  df-plusg 14256  df-mulr 14257  df-starv 14258  df-sca 14259  df-vsca 14260  df-ip 14261  df-tset 14262  df-ple 14263  df-ds 14265  df-unif 14266  df-hom 14267  df-cco 14268  df-0g 14385  df-gsum 14386  df-prds 14391  df-pws 14393  df-mre 14529  df-mrc 14530  df-acs 14532  df-mnd 15420  df-mhm 15469  df-submnd 15470  df-grp 15550  df-minusg 15551  df-sbg 15552  df-mulg 15553  df-subg 15683  df-ghm 15750  df-cntz 15840  df-cmn 16284  df-abl 16285  df-mgp 16597  df-ur 16609  df-srg 16613  df-rng 16652  df-cring 16653  df-oppr 16720  df-dvdsr 16738  df-unit 16739  df-invr 16769  df-rnghom 16811  df-subrg 16868  df-lmod 16955  df-lss 17019  df-lsp 17058  df-nzr 17345  df-rlreg 17359  df-domn 17360  df-idom 17361  df-assa 17389  df-asp 17390  df-ascl 17391  df-psr 17428  df-mvr 17429  df-mpl 17430  df-opsr 17432  df-evls 17593  df-evl 17594  df-psr1 17641  df-vr1 17642  df-ply1 17643  df-coe1 17644  df-evl1 17756  df-cnfld 17824  df-mdeg 21529  df-deg1 21530  df-mon1 21607  df-uc1p 21608  df-q1p 21609  df-r1p 21610
This theorem is referenced by:  fta1glem2  21643
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