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Theorem vieta1lem1 22872
Description: Lemma for vieta1 22874. (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 22763 . . . . 5  |-  (Poly `  S )  C_  (Poly `  CC )
3 vieta1.4 . . . . . 6  |-  ( ph  ->  F  e.  (Poly `  S ) )
43adantr 463 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  S )
)
52, 4sseldi 3487 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  CC )
)
6 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
7 cnvimass 5345 . . . . . . . . 9  |-  ( `' F " { 0 } )  C_  dom  F
86, 7eqsstri 3519 . . . . . . . 8  |-  R  C_  dom  F
9 plyf 22761 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
103, 9syl 16 . . . . . . . . 9  |-  ( ph  ->  F : CC --> CC )
11 fdm 5717 . . . . . . . . 9  |-  ( F : CC --> CC  ->  dom 
F  =  CC )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  F  =  CC )
138, 12syl5sseq 3537 . . . . . . 7  |-  ( ph  ->  R  C_  CC )
1413sselda 3489 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  z  e.  CC )
15 eqid 2454 . . . . . . 7  |-  ( Xp  oF  -  ( CC  X.  { z } ) )  =  ( Xp  oF  -  ( CC 
X.  { z } ) )
1615plyremlem 22866 . . . . . 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 1006 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC ) )
1917simp2d 1007 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1 )
20 ax-1ne0 9550 . . . . . . 7  |-  1  =/=  0
2120a1i 11 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  1  =/=  0 )
2219, 21eqnetrd 2747 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =/=  0 )
23 fveq2 5848 . . . . . . 7  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  (deg
`  0p ) )
24 dgr0 22825 . . . . . . 7  |-  (deg ` 
0p )  =  0
2523, 24syl6eq 2511 . . . . . 6  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  0 )
2625necon3i 2694 . . . . 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 22864 . . . 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 1226 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  e.  (Poly `  CC )
)
301, 29syl5eqel 2546 . 2  |-  ( (
ph  /\  z  e.  R )  ->  Q  e.  (Poly `  CC )
)
31 ax-1cn 9539 . . . 4  |-  1  e.  CC
3231a1i 11 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  1  e.  CC )
33 vieta1lem.6 . . . . 5  |-  ( ph  ->  D  e.  NN )
3433nncnd 10547 . . . 4  |-  ( ph  ->  D  e.  CC )
3534adantr 463 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  CC )
36 dgrcl 22796 . . . . 5  |-  ( Q  e.  (Poly `  CC )  ->  (deg `  Q
)  e.  NN0 )
3730, 36syl 16 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  NN0 )
3837nn0cnd 10850 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  CC )
39 addcom 9755 . . . . 5  |-  ( ( 1  e.  CC  /\  D  e.  CC )  ->  ( 1  +  D
)  =  ( D  +  1 ) )
4031, 35, 39sylancr 661 . . . 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 2511 . . . . . 6  |-  ( ph  ->  ( D  +  1 )  =  (deg `  F ) )
4443adantr 463 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =  (deg `  F
) )
456eleq2i 2532 . . . . . . . . . 10  |-  ( z  e.  R  <->  z  e.  ( `' F " { 0 } ) )
46 ffn 5713 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
4710, 46syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
48 fniniseg 5984 . . . . . . . . . . 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 622 . . . . . . . 8  |-  ( (
ph  /\  z  e.  R )  ->  ( F `  z )  =  0 )
5215facth 22868 . . . . . . . 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 1226 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
541oveq2i 6281 . . . . . . 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 2513 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )
5655fveq2d 5852 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  (deg
`  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
5733peano2nnd 10548 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D  +  1 )  e.  NN )
5841, 57eqeltrrd 2543 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN )
5958nnne0d 10576 . . . . . . . . . . . . 13  |-  ( ph  ->  N  =/=  0 )
6042, 59syl5eqner 2755 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  F )  =/=  0 )
61 fveq2 5848 . . . . . . . . . . . . . 14  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
6261, 24syl6eq 2511 . . . . . . . . . . . . 13  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
6362necon3i 2694 . . . . . . . . . . . 12  |-  ( (deg
`  F )  =/=  0  ->  F  =/=  0p )
6460, 63syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  =/=  0p )
6564adantr 463 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  F  =/=  0p )
6655, 65eqnetrrd 2748 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =/=  0p )
67 plymul0or 22843 . . . . . . . . . . 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 659 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  (
( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q )  =  0p  <->  ( (
Xp  oF  -  ( CC  X.  { z } ) )  =  0p  \/  Q  =  0p ) ) )
6968necon3abid 2700 . . . . . . . . 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 2779 . . . . . . . 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 461 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  Q  =/=  0p )
74 eqid 2454 . . . . . . 7  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
75 eqid 2454 . . . . . . 7  |-  (deg `  Q )  =  (deg
`  Q )
7674, 75dgrmul 22833 . . . . . 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 1227 . . . . 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 2499 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
7919oveq1d 6285 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
(deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
)  =  ( 1  +  (deg `  Q
) ) )
8040, 78, 793eqtrd 2499 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( 1  +  (deg `  Q )
) )
8132, 35, 38, 80addcanad 9774 . 2  |-  ( (
ph  /\  z  e.  R )  ->  D  =  (deg `  Q )
)
8230, 81jca 530 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   {csn 4016    X. cxp 4986   `'ccnv 4987   dom cdm 4988   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    oFcof 6511   CCcc 9479   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486    - cmin 9796   -ucneg 9797    / cdiv 10202   NNcn 10531   NN0cn0 10791   #chash 12387   sum_csu 13590   0pc0p 22242  Polycply 22747   Xpcidp 22748  coeffccoe 22749  degcdgr 22750   quot cquot 22852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12090  df-exp 12149  df-hash 12388  df-cj 13014  df-re 13015  df-im 13016  df-sqrt 13150  df-abs 13151  df-clim 13393  df-rlim 13394  df-sum 13591  df-0p 22243  df-ply 22751  df-idp 22752  df-coe 22753  df-dgr 22754  df-quot 22853
This theorem is referenced by:  vieta1lem2  22873
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