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Theorem plydivlem3 20165
Description: Lemma for plydivex 20167. 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 20066 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
3 ply0 20080 . . 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 9027 . . . . . . 7  |-  CC  e.  _V
76a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
8 plyf 20070 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
9 ffn 5550 . . . . . . 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 20070 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
13 ffn 5550 . . . . . . . 8  |-  ( G : CC --> CC  ->  G  Fn  CC )
1411, 12, 133syl 19 . . . . . . 7  |-  ( ph  ->  G  Fn  CC )
15 plyf 20070 . . . . . . . 8  |-  ( 0 p  e.  (Poly `  S )  ->  0 p : CC --> CC )
16 ffn 5550 . . . . . . . 8  |-  ( 0 p : CC --> CC  ->  0 p  Fn  CC )
174, 15, 163syl 19 . . . . . . 7  |-  ( ph  ->  0 p  Fn  CC )
18 inidm 3510 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
1914, 17, 7, 7, 18offn 6275 . . . . . 6  |-  ( ph  ->  ( G  o F  x.  0 p )  Fn  CC )
20 eqidd 2405 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  =  ( F `  z
) )
21 eqidd 2405 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  =  ( G `  z
) )
22 0pval 19516 . . . . . . . . 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 6273 . . . . . . 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 5829 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
2726mul01d 9221 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z )  x.  0 )  =  0 )
2824, 27eqtrd 2436 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  0 p ) `  z
)  =  0 )
291, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
3029ffvelrnda 5829 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
3130subid1d 9356 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  0 )  =  ( F `  z
) )
327, 10, 19, 10, 20, 28, 31offveq 6284 . . . . 5  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  0 p
) )  =  F )
3332eqeq1d 2412 . . . 4  |-  ( ph  ->  ( ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p  <->  F  = 
0 p ) )
3432fveq2d 5691 . . . . . 6  |-  ( ph  ->  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  =  (deg `  F
) )
35 dgrcl 20105 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3611, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3736nn0red 10231 . . . . . . . . 9  |-  ( ph  ->  (deg `  G )  e.  RR )
3837recnd 9070 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  CC )
3938addid2d 9223 . . . . . . 7  |-  ( ph  ->  ( 0  +  (deg
`  G ) )  =  (deg `  G
) )
4039eqcomd 2409 . . . . . 6  |-  ( ph  ->  (deg `  G )  =  ( 0  +  (deg `  G )
) )
4134, 40breq12d 4185 . . . . 5  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  <  (deg `  G
)  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
42 dgrcl 20105 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
431, 42syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  F )  e.  NN0 )
4443nn0red 10231 . . . . . 6  |-  ( ph  ->  (deg `  F )  e.  RR )
45 0re 9047 . . . . . . 7  |-  0  e.  RR
4645a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR )
4744, 37, 46ltsubaddd 9578 . . . . 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 6048 . . . . . . 7  |-  ( q  =  0 p  -> 
( G  o F  x.  q )  =  ( G  o F  x.  0 p ) )
5352oveq2d 6056 . . . . . 6  |-  ( q  =  0 p  -> 
( F  o F  -  ( G  o F  x.  q )
)  =  ( F  o F  -  ( G  o F  x.  0 p ) ) )
5451, 53syl5eq 2448 . . . . 5  |-  ( q  =  0 p  ->  R  =  ( F  o F  -  ( G  o F  x.  0 p ) ) )
5554eqeq1d 2412 . . . 4  |-  ( q  =  0 p  -> 
( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p ) )
5654fveq2d 5691 . . . . 5  |-  ( q  =  0 p  -> 
(deg `  R )  =  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) ) )
5756breq1d 4182 . . . 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 3012 . 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 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   _Vcvv 2916    C_ wss 3280   class class class wbr 4172    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   RRcr 8945   0cc0 8946   1c1 8947    + caddc 8949    x. cmul 8951    < clt 9076    - cmin 9247   -ucneg 9248    / cdiv 9633   NN0cn0 10177   0 pc0p 19514  Polycply 20056  degcdgr 20059
This theorem is referenced by:  plydivex  20167
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-rlim 12238  df-sum 12435  df-0p 19515  df-ply 20060  df-coe 20062  df-dgr 20063
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