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Theorem plydivlem3 20204
Description: Lemma for plydivex 20206. Base case of induction. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
plydiv.pl  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  +  y )  e.  S )
plydiv.tm  |-  ( (
ph  /\  ( x  e.  S  /\  y  e.  S ) )  -> 
( x  x.  y
)  e.  S )
plydiv.rc  |-  ( (
ph  /\  ( x  e.  S  /\  x  =/=  0 ) )  -> 
( 1  /  x
)  e.  S )
plydiv.m1  |-  ( ph  -> 
-u 1  e.  S
)
plydiv.f  |-  ( ph  ->  F  e.  (Poly `  S ) )
plydiv.g  |-  ( ph  ->  G  e.  (Poly `  S ) )
plydiv.z  |-  ( ph  ->  G  =/=  0 p )
plydiv.r  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
plydiv.0  |-  ( ph  ->  ( F  =  0 p  \/  ( (deg
`  F )  -  (deg `  G ) )  <  0 ) )
Assertion
Ref Expression
plydivlem3  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
Distinct variable groups:    x, y,
q, F    ph, x, y    G, q, x, y    x, R, y    S, q, x, y
Allowed substitution hints:    ph( q)    R( q)

Proof of Theorem plydivlem3
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 plydiv.f . . 3  |-  ( ph  ->  F  e.  (Poly `  S ) )
2 plybss 20105 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
3 ply0 20119 . . 3  |-  ( S 
C_  CC  ->  0 p  e.  (Poly `  S
) )
41, 2, 33syl 19 . 2  |-  ( ph  ->  0 p  e.  (Poly `  S ) )
5 plydiv.0 . . 3  |-  ( ph  ->  ( F  =  0 p  \/  ( (deg
`  F )  -  (deg `  G ) )  <  0 ) )
6 cnex 9063 . . . . . . 7  |-  CC  e.  _V
76a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
8 plyf 20109 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
9 ffn 5583 . . . . . . 7  |-  ( F : CC --> CC  ->  F  Fn  CC )
101, 8, 93syl 19 . . . . . 6  |-  ( ph  ->  F  Fn  CC )
11 plydiv.g . . . . . . . 8  |-  ( ph  ->  G  e.  (Poly `  S ) )
12 plyf 20109 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
13 ffn 5583 . . . . . . . 8  |-  ( G : CC --> CC  ->  G  Fn  CC )
1411, 12, 133syl 19 . . . . . . 7  |-  ( ph  ->  G  Fn  CC )
15 plyf 20109 . . . . . . . 8  |-  ( 0 p  e.  (Poly `  S )  ->  0 p : CC --> CC )
16 ffn 5583 . . . . . . . 8  |-  ( 0 p : CC --> CC  ->  0 p  Fn  CC )
174, 15, 163syl 19 . . . . . . 7  |-  ( ph  ->  0 p  Fn  CC )
18 inidm 3542 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
1914, 17, 7, 7, 18offn 6308 . . . . . 6  |-  ( ph  ->  ( G  o F  x.  0 p )  Fn  CC )
20 eqidd 2436 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  =  ( F `  z
) )
21 eqidd 2436 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  =  ( G `  z
) )
22 0pval 19555 . . . . . . . . 9  |-  ( z  e.  CC  ->  (
0 p `  z
)  =  0 )
2322adantl 453 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0 p `  z )  =  0 )
2414, 17, 7, 7, 18, 21, 23ofval 6306 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  0 p ) `  z
)  =  ( ( G `  z )  x.  0 ) )
2511, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  G : CC --> CC )
2625ffvelrnda 5862 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
2726mul01d 9257 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z )  x.  0 )  =  0 )
2824, 27eqtrd 2467 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  0 p ) `  z
)  =  0 )
291, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
3029ffvelrnda 5862 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
3130subid1d 9392 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  0 )  =  ( F `  z
) )
327, 10, 19, 10, 20, 28, 31offveq 6317 . . . . 5  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  0 p
) )  =  F )
3332eqeq1d 2443 . . . 4  |-  ( ph  ->  ( ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p  <->  F  = 
0 p ) )
3432fveq2d 5724 . . . . . 6  |-  ( ph  ->  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  =  (deg `  F
) )
35 dgrcl 20144 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3611, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3736nn0red 10267 . . . . . . . . 9  |-  ( ph  ->  (deg `  G )  e.  RR )
3837recnd 9106 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  CC )
3938addid2d 9259 . . . . . . 7  |-  ( ph  ->  ( 0  +  (deg
`  G ) )  =  (deg `  G
) )
4039eqcomd 2440 . . . . . 6  |-  ( ph  ->  (deg `  G )  =  ( 0  +  (deg `  G )
) )
4134, 40breq12d 4217 . . . . 5  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  <  (deg `  G
)  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
42 dgrcl 20144 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
431, 42syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  F )  e.  NN0 )
4443nn0red 10267 . . . . . 6  |-  ( ph  ->  (deg `  F )  e.  RR )
45 0re 9083 . . . . . . 7  |-  0  e.  RR
4645a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR )
4744, 37, 46ltsubaddd 9614 . . . . 5  |-  ( ph  ->  ( ( (deg `  F )  -  (deg `  G ) )  <  0  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
4841, 47bitr4d 248 . . . 4  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  <  (deg `  G
)  <->  ( (deg `  F )  -  (deg `  G ) )  <  0 ) )
4933, 48orbi12d 691 . . 3  |-  ( ph  ->  ( ( ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  0 p
) ) )  < 
(deg `  G )
)  <->  ( F  =  0 p  \/  (
(deg `  F )  -  (deg `  G )
)  <  0 ) ) )
505, 49mpbird 224 . 2  |-  ( ph  ->  ( ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  0 p
) ) )  < 
(deg `  G )
) )
51 plydiv.r . . . . . 6  |-  R  =  ( F  o F  -  ( G  o F  x.  q )
)
52 oveq2 6081 . . . . . . 7  |-  ( q  =  0 p  -> 
( G  o F  x.  q )  =  ( G  o F  x.  0 p ) )
5352oveq2d 6089 . . . . . 6  |-  ( q  =  0 p  -> 
( F  o F  -  ( G  o F  x.  q )
)  =  ( F  o F  -  ( G  o F  x.  0 p ) ) )
5451, 53syl5eq 2479 . . . . 5  |-  ( q  =  0 p  ->  R  =  ( F  o F  -  ( G  o F  x.  0 p ) ) )
5554eqeq1d 2443 . . . 4  |-  ( q  =  0 p  -> 
( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p ) )
5654fveq2d 5724 . . . . 5  |-  ( q  =  0 p  -> 
(deg `  R )  =  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) ) )
5756breq1d 4214 . . . 4  |-  ( q  =  0 p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  <  (deg `  G
) ) )
5855, 57orbi12d 691 . . 3  |-  ( q  =  0 p  -> 
( ( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  0 p
) ) )  < 
(deg `  G )
) ) )
5958rspcev 3044 . 2  |-  ( ( 0 p  e.  (Poly `  S )  /\  (
( F  o F  -  ( G  o F  x.  0 p
) )  =  0 p  \/  (deg `  ( F  o F  -  ( G  o F  x.  0 p
) ) )  < 
(deg `  G )
) )  ->  E. q  e.  (Poly `  S )
( R  =  0 p  \/  (deg `  R )  <  (deg `  G ) ) )
604, 50, 59syl2anc 643 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0 p  \/  (deg `  R )  < 
(deg `  G )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   E.wrex 2698   _Vcvv 2948    C_ wss 3312   class class class wbr 4204    Fn wfn 5441   -->wf 5442   ` cfv 5446  (class class class)co 6073    o Fcof 6295   CCcc 8980   RRcr 8981   0cc0 8982   1c1 8983    + caddc 8985    x. cmul 8987    < clt 9112    - cmin 9283   -ucneg 9284    / cdiv 9669   NN0cn0 10213   0 pc0p 19553  Polycply 20095  degcdgr 20098
This theorem is referenced by:  plydivex  20206
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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