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Theorem vieta1lem1 22456
Description: Lemma for vieta1 22458. (Contributed by Mario Carneiro, 28-Jul-2014.)
Hypotheses
Ref Expression
vieta1.1  |-  A  =  (coeff `  F )
vieta1.2  |-  N  =  (deg `  F )
vieta1.3  |-  R  =  ( `' F " { 0 } )
vieta1.4  |-  ( ph  ->  F  e.  (Poly `  S ) )
vieta1.5  |-  ( ph  ->  ( # `  R
)  =  N )
vieta1lem.6  |-  ( ph  ->  D  e.  NN )
vieta1lem.7  |-  ( ph  ->  ( D  +  1 )  =  N )
vieta1lem.8  |-  ( ph  ->  A. f  e.  (Poly `  CC ) ( ( D  =  (deg `  f )  /\  ( # `
 ( `' f
" { 0 } ) )  =  (deg
`  f ) )  ->  sum_ x  e.  ( `' f " {
0 } ) x  =  -u ( ( (coeff `  f ) `  (
(deg `  f )  -  1 ) )  /  ( (coeff `  f ) `  (deg `  f ) ) ) ) )
vieta1lem.9  |-  Q  =  ( F quot  ( Xp  oF  -  ( CC  X.  { z } ) ) )
Assertion
Ref Expression
vieta1lem1  |-  ( (
ph  /\  z  e.  R )  ->  ( Q  e.  (Poly `  CC )  /\  D  =  (deg
`  Q ) ) )
Distinct variable groups:    D, f    f, F    z, f, N   
x, f, Q    R, f    x, z, R    A, f, z    ph, x, z
Allowed substitution hints:    ph( f)    A( x)    D( x, z)    Q( z)    S( x, z, f)    F( x, z)    N( x)

Proof of Theorem vieta1lem1
StepHypRef Expression
1 vieta1lem.9 . . 3  |-  Q  =  ( F quot  ( Xp  oF  -  ( CC  X.  { z } ) ) )
2 plyssc 22348 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
3 vieta1.4 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  S ) )
43adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  S )
)
52, 4sseldi 3502 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  CC )
)
6 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
7 cnvimass 5356 . . . . . . . . 9  |-  ( `' F " { 0 } )  C_  dom  F
86, 7eqsstri 3534 . . . . . . . 8  |-  R  C_  dom  F
9 plyf 22346 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
103, 9syl 16 . . . . . . . . 9  |-  ( ph  ->  F : CC --> CC )
11 fdm 5734 . . . . . . . . 9  |-  ( F : CC --> CC  ->  dom 
F  =  CC )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  F  =  CC )
138, 12syl5sseq 3552 . . . . . . 7  |-  ( ph  ->  R  C_  CC )
1413sselda 3504 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  z  e.  CC )
15 eqid 2467 . . . . . . 7  |-  ( Xp  oF  -  ( CC  X.  { z } ) )  =  ( Xp  oF  -  ( CC 
X.  { z } ) )
1615plyremlem 22450 . . . . . 6  |-  ( z  e.  CC  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { z } ) ) " {
0 } )  =  { z } ) )
1714, 16syl 16 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1  /\  ( `' ( Xp  oF  -  ( CC 
X.  { z } ) ) " {
0 } )  =  { z } ) )
1817simp1d 1008 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC ) )
1917simp2d 1009 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1 )
20 ax-1ne0 9560 . . . . . . 7  |-  1  =/=  0
2120a1i 11 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  1  =/=  0 )
2219, 21eqnetrd 2760 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =/=  0 )
23 fveq2 5865 . . . . . . 7  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  (deg
`  0p ) )
24 dgr0 22409 . . . . . . 7  |-  (deg ` 
0p )  =  0
2523, 24syl6eq 2524 . . . . . 6  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  0 )
2625necon3i 2707 . . . . 5  |-  ( (deg
`  ( Xp  oF  -  ( CC  X.  { z } ) ) )  =/=  0  ->  ( Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )
2722, 26syl 16 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )
28 quotcl2 22448 . . . 4  |-  ( ( F  e.  (Poly `  CC )  /\  (
Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  (
Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )  ->  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  e.  (Poly `  CC )
)
295, 18, 27, 28syl3anc 1228 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  e.  (Poly `  CC )
)
301, 29syl5eqel 2559 . 2  |-  ( (
ph  /\  z  e.  R )  ->  Q  e.  (Poly `  CC )
)
31 ax-1cn 9549 . . . 4  |-  1  e.  CC
3231a1i 11 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  1  e.  CC )
33 vieta1lem.6 . . . . 5  |-  ( ph  ->  D  e.  NN )
3433nncnd 10551 . . . 4  |-  ( ph  ->  D  e.  CC )
3534adantr 465 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  CC )
36 dgrcl 22381 . . . . 5  |-  ( Q  e.  (Poly `  CC )  ->  (deg `  Q
)  e.  NN0 )
3730, 36syl 16 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  NN0 )
3837nn0cnd 10853 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  CC )
39 addcom 9764 . . . . 5  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
4031, 35, 39sylancr 663 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( D  + 
1 ) )
41 vieta1lem.7 . . . . . . 7  |-  ( ph  ->  ( D  +  1 )  =  N )
42 vieta1.2 . . . . . . 7  |-  N  =  (deg `  F )
4341, 42syl6eq 2524 . . . . . 6  |-  ( ph  ->  ( D  +  1 )  =  (deg `  F ) )
4443adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =  (deg `  F
) )
456eleq2i 2545 . . . . . . . . . 10  |-  ( z  e.  R  <->  z  e.  ( `' F " { 0 } ) )
46 ffn 5730 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
4710, 46syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
48 fniniseg 6001 . . . . . . . . . . 11  |-  ( F  Fn  CC  ->  (
z  e.  ( `' F " { 0 } )  <->  ( z  e.  CC  /\  ( F `
 z )  =  0 ) ) )
4947, 48syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( z  e.  ( `' F " { 0 } )  <->  ( z  e.  CC  /\  ( F `
 z )  =  0 ) ) )
5045, 49syl5bb 257 . . . . . . . . 9  |-  ( ph  ->  ( z  e.  R  <->  ( z  e.  CC  /\  ( F `  z )  =  0 ) ) )
5150simplbda 624 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( F `  z )  =  0 )
5215facth 22452 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC  /\  ( F `
 z )  =  0 )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
534, 14, 51, 52syl3anc 1228 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
541oveq2i 6294 . . . . . . 7  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =  ( ( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  ( F quot  (
Xp  oF  -  ( CC  X.  { z } ) ) ) )
5553, 54syl6eqr 2526 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )
5655fveq2d 5869 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  (deg
`  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
5733peano2nnd 10552 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D  +  1 )  e.  NN )
5841, 57eqeltrrd 2556 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN )
5958nnne0d 10579 . . . . . . . . . . . . 13  |-  ( ph  ->  N  =/=  0 )
6042, 59syl5eqner 2768 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  F )  =/=  0 )
61 fveq2 5865 . . . . . . . . . . . . . 14  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
6261, 24syl6eq 2524 . . . . . . . . . . . . 13  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
6362necon3i 2707 . . . . . . . . . . . 12  |-  ( (deg
`  F )  =/=  0  ->  F  =/=  0p )
6460, 63syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  =/=  0p )
6564adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  F  =/=  0p )
6655, 65eqnetrrd 2761 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =/=  0p )
67 plymul0or 22427 . . . . . . . . . . 11  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  e.  (Poly `  CC )  /\  Q  e.  (Poly `  CC )
)  ->  ( (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =  0p  <->  ( (
Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
6818, 30, 67syl2anc 661 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =  0p  <->  ( (
Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
6968necon3abid 2713 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =/=  0p  <->  -.  (
( Xp  oF  -  ( CC 
X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
7066, 69mpbid 210 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  -.  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) )
71 neanior 2792 . . . . . . . 8  |-  ( ( ( Xp  oF  -  ( CC 
X.  { z } ) )  =/=  0p  /\  Q  =/=  0p )  <->  -.  (
( Xp  oF  -  ( CC 
X.  { z } ) )  =  0p  \/  Q  =  0p ) )
7270, 71sylibr 212 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  =/=  0p  /\  Q  =/=  0p ) )
7372simprd 463 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  Q  =/=  0p )
74 eqid 2467 . . . . . . 7  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
75 eqid 2467 . . . . . . 7  |-  (deg `  Q )  =  (deg
`  Q )
7674, 75dgrmul 22417 . . . . . 6  |-  ( ( ( ( Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC )  /\  (
Xp  oF  -  ( CC  X.  { z } ) )  =/=  0p )  /\  ( Q  e.  (Poly `  CC )  /\  Q  =/=  0p ) )  -> 
(deg `  ( (
Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )  =  ( (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg
`  Q ) ) )
7718, 27, 30, 73, 76syl22anc 1229 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
7844, 56, 773eqtrd 2512 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
7919oveq1d 6298 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
(deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
)  =  ( 1  +  (deg `  Q
) ) )
8040, 78, 793eqtrd 2512 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( 1  +  (deg `  Q )
) )
8132, 35, 38, 80addcanad 9783 . 2  |-  ( (
ph  /\  z  e.  R )  ->  D  =  (deg `  Q )
)
8230, 81jca 532 1  |-  ( (
ph  /\  z  e.  R )  ->  ( Q  e.  (Poly `  CC )  /\  D  =  (deg
`  Q ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   {csn 4027    X. cxp 4997   `'ccnv 4998   dom cdm 4999   "cima 5002    Fn wfn 5582   -->wf 5583   ` cfv 5587  (class class class)co 6283    oFcof 6521   CCcc 9489   0cc0 9491   1c1 9492    + caddc 9494    x. cmul 9496    - cmin 9804   -ucneg 9805    / cdiv 10205   NNcn 10535   NN0cn0 10794   #chash 12372   sum_csu 13470   0pc0p 21827  Polycply 22332   Xpcidp 22333  coeffccoe 22334  degcdgr 22335   quot cquot 22436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-inf2 8057  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569  ax-addf 9570
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-isom 5596  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-of 6523  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7900  df-oi 7934  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-div 10206  df-nn 10536  df-2 10593  df-3 10594  df-n0 10795  df-z 10864  df-uz 11082  df-rp 11220  df-fz 11672  df-fzo 11792  df-fl 11896  df-seq 12075  df-exp 12134  df-hash 12373  df-cj 12894  df-re 12895  df-im 12896  df-sqrt 13030  df-abs 13031  df-clim 13273  df-rlim 13274  df-sum 13471  df-0p 21828  df-ply 22336  df-idp 22337  df-coe 22338  df-dgr 22339  df-quot 22437
This theorem is referenced by:  vieta1lem2  22457
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