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Theorem vieta1lem1 21788
Description: Lemma for vieta1 21790. (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 21680 . . . . 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 3366 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  F  e.  (Poly `  CC )
)
6 vieta1.3 . . . . . . . . 9  |-  R  =  ( `' F " { 0 } )
7 cnvimass 5201 . . . . . . . . 9  |-  ( `' F " { 0 } )  C_  dom  F
86, 7eqsstri 3398 . . . . . . . 8  |-  R  C_  dom  F
9 plyf 21678 . . . . . . . . . 10  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
103, 9syl 16 . . . . . . . . 9  |-  ( ph  ->  F : CC --> CC )
11 fdm 5575 . . . . . . . . 9  |-  ( F : CC --> CC  ->  dom 
F  =  CC )
1210, 11syl 16 . . . . . . . 8  |-  ( ph  ->  dom  F  =  CC )
138, 12syl5sseq 3416 . . . . . . 7  |-  ( ph  ->  R  C_  CC )
1413sselda 3368 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  z  e.  CC )
15 eqid 2443 . . . . . . 7  |-  ( Xp  oF  -  ( CC  X.  { z } ) )  =  ( Xp  oF  -  ( CC 
X.  { z } ) )
1615plyremlem 21782 . . . . . 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 1000 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
Xp  oF  -  ( CC  X.  { z } ) )  e.  (Poly `  CC ) )
1917simp2d 1001 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  1 )
20 ax-1ne0 9363 . . . . . . 7  |-  1  =/=  0
2120a1i 11 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  1  =/=  0 )
2219, 21eqnetrd 2638 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =/=  0 )
23 fveq2 5703 . . . . . . 7  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  (deg
`  0p ) )
24 dgr0 21741 . . . . . . 7  |-  (deg ` 
0p )  =  0
2523, 24syl6eq 2491 . . . . . 6  |-  ( ( Xp  oF  -  ( CC  X.  { z } ) )  =  0p  ->  (deg `  (
Xp  oF  -  ( CC  X.  { z } ) ) )  =  0 )
2625necon3i 2662 . . . . 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 21780 . . . 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 1218 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  e.  (Poly `  CC )
)
301, 29syl5eqel 2527 . 2  |-  ( (
ph  /\  z  e.  R )  ->  Q  e.  (Poly `  CC )
)
31 ax-1cn 9352 . . . 4  |-  1  e.  CC
3231a1i 11 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  1  e.  CC )
33 vieta1lem.6 . . . . 5  |-  ( ph  ->  D  e.  NN )
3433nncnd 10350 . . . 4  |-  ( ph  ->  D  e.  CC )
3534adantr 465 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  D  e.  CC )
36 dgrcl 21713 . . . . 5  |-  ( Q  e.  (Poly `  CC )  ->  (deg `  Q
)  e.  NN0 )
3730, 36syl 16 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  NN0 )
3837nn0cnd 10650 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  Q )  e.  CC )
39 addcom 9567 . . . . 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 2491 . . . . . 6  |-  ( ph  ->  ( D  +  1 )  =  (deg `  F ) )
4443adantr 465 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =  (deg `  F
) )
456eleq2i 2507 . . . . . . . . . 10  |-  ( z  e.  R  <->  z  e.  ( `' F " { 0 } ) )
46 ffn 5571 . . . . . . . . . . . 12  |-  ( F : CC --> CC  ->  F  Fn  CC )
4710, 46syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  Fn  CC )
48 fniniseg 5836 . . . . . . . . . . 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 21784 . . . . . . . 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 1218 . . . . . . 7  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  ( F quot  ( Xp  oF  -  ( CC 
X.  { z } ) ) ) ) )
541oveq2i 6114 . . . . . . 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 2493 . . . . . 6  |-  ( (
ph  /\  z  e.  R )  ->  F  =  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) )
5655fveq2d 5707 . . . . 5  |-  ( (
ph  /\  z  e.  R )  ->  (deg `  F )  =  (deg
`  ( ( Xp  oF  -  ( CC  X.  { z } ) )  oF  x.  Q ) ) )
5733peano2nnd 10351 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( D  +  1 )  e.  NN )
5841, 57eqeltrrd 2518 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN )
5958nnne0d 10378 . . . . . . . . . . . . 13  |-  ( ph  ->  N  =/=  0 )
6042, 59syl5eqner 2645 . . . . . . . . . . . 12  |-  ( ph  ->  (deg `  F )  =/=  0 )
61 fveq2 5703 . . . . . . . . . . . . . 14  |-  ( F  =  0p  -> 
(deg `  F )  =  (deg `  0p
) )
6261, 24syl6eq 2491 . . . . . . . . . . . . 13  |-  ( F  =  0p  -> 
(deg `  F )  =  0 )
6362necon3i 2662 . . . . . . . . . . . 12  |-  ( (deg
`  F )  =/=  0  ->  F  =/=  0p )
6460, 63syl 16 . . . . . . . . . . 11  |-  ( ph  ->  F  =/=  0p )
6564adantr 465 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  R )  ->  F  =/=  0p )
6655, 65eqnetrrd 2640 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  R )  ->  (
( Xp  oF  -  ( CC 
X.  { z } ) )  oF  x.  Q )  =/=  0p )
67 plymul0or 21759 . . . . . . . . . . 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 2653 . . . . . . . . 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 2709 . . . . . . . 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 2443 . . . . . . 7  |-  (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  =  (deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )
75 eqid 2443 . . . . . . 7  |-  (deg `  Q )  =  (deg
`  Q )
7674, 75dgrmul 21749 . . . . . 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 1219 . . . . 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 2479 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  ( D  +  1 )  =  ( (deg `  ( Xp  oF  -  ( CC 
X.  { z } ) ) )  +  (deg `  Q )
) )
7919oveq1d 6118 . . . 4  |-  ( (
ph  /\  z  e.  R )  ->  (
(deg `  ( Xp  oF  -  ( CC  X.  { z } ) ) )  +  (deg `  Q )
)  =  ( 1  +  (deg `  Q
) ) )
8040, 78, 793eqtrd 2479 . . 3  |-  ( (
ph  /\  z  e.  R )  ->  (
1  +  D )  =  ( 1  +  (deg `  Q )
) )
8132, 35, 38, 80addcanad 9586 . 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 965    = wceq 1369    e. wcel 1756    =/= wne 2618   A.wral 2727   {csn 3889    X. cxp 4850   `'ccnv 4851   dom cdm 4852   "cima 4855    Fn wfn 5425   -->wf 5426   ` cfv 5430  (class class class)co 6103    oFcof 6330   CCcc 9292   0cc0 9294   1c1 9295    + caddc 9297    x. cmul 9299    - cmin 9607   -ucneg 9608    / cdiv 10005   NNcn 10334   NN0cn0 10591   #chash 12115   sum_csu 13175   0pc0p 21159  Polycply 21664   Xpcidp 21665  coeffccoe 21666  degcdgr 21667   quot cquot 21768
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 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-inf2 7859  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372  ax-addf 9373
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-se 4692  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-of 6332  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-map 7228  df-pm 7229  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-oi 7736  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-3 10393  df-n0 10592  df-z 10659  df-uz 10874  df-rp 11004  df-fz 11450  df-fzo 11561  df-fl 11654  df-seq 11819  df-exp 11878  df-hash 12116  df-cj 12600  df-re 12601  df-im 12602  df-sqr 12736  df-abs 12737  df-clim 12978  df-rlim 12979  df-sum 13176  df-0p 21160  df-ply 21668  df-idp 21669  df-coe 21670  df-dgr 21671  df-quot 21769
This theorem is referenced by:  vieta1lem2  21789
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