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Theorem plydivlem3 20079
Description: Lemma for plydivex 20081. 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 19980 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
3 ply0 19994 . . 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 9004 . . . . . . 7  |-  CC  e.  _V
76a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
8 plyf 19984 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
9 ffn 5531 . . . . . . 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 19984 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
13 ffn 5531 . . . . . . . 8  |-  ( G : CC --> CC  ->  G  Fn  CC )
1411, 12, 133syl 19 . . . . . . 7  |-  ( ph  ->  G  Fn  CC )
15 plyf 19984 . . . . . . . 8  |-  ( 0 p  e.  (Poly `  S )  ->  0 p : CC --> CC )
16 ffn 5531 . . . . . . . 8  |-  ( 0 p : CC --> CC  ->  0 p  Fn  CC )
174, 15, 163syl 19 . . . . . . 7  |-  ( ph  ->  0 p  Fn  CC )
18 inidm 3493 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
1914, 17, 7, 7, 18offn 6255 . . . . . 6  |-  ( ph  ->  ( G  o F  x.  0 p )  Fn  CC )
20 eqidd 2388 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  =  ( F `  z
) )
21 eqidd 2388 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  =  ( G `  z
) )
22 0pval 19430 . . . . . . . . 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 6253 . . . . . . 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 5809 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
2726mul01d 9197 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z )  x.  0 )  =  0 )
2824, 27eqtrd 2419 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  o F  x.  0 p ) `  z
)  =  0 )
291, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
3029ffvelrnda 5809 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
3130subid1d 9332 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  0 )  =  ( F `  z
) )
327, 10, 19, 10, 20, 28, 31offveq 6264 . . . . 5  |-  ( ph  ->  ( F  o F  -  ( G  o F  x.  0 p
) )  =  F )
3332eqeq1d 2395 . . . 4  |-  ( ph  ->  ( ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p  <->  F  = 
0 p ) )
3432fveq2d 5672 . . . . . 6  |-  ( ph  ->  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  =  (deg `  F
) )
35 dgrcl 20019 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3611, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3736nn0red 10207 . . . . . . . . 9  |-  ( ph  ->  (deg `  G )  e.  RR )
3837recnd 9047 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  CC )
3938addid2d 9199 . . . . . . 7  |-  ( ph  ->  ( 0  +  (deg
`  G ) )  =  (deg `  G
) )
4039eqcomd 2392 . . . . . 6  |-  ( ph  ->  (deg `  G )  =  ( 0  +  (deg `  G )
) )
4134, 40breq12d 4166 . . . . 5  |-  ( ph  ->  ( (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) )  <  (deg `  G
)  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
42 dgrcl 20019 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
431, 42syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  F )  e.  NN0 )
4443nn0red 10207 . . . . . 6  |-  ( ph  ->  (deg `  F )  e.  RR )
45 0re 9024 . . . . . . 7  |-  0  e.  RR
4645a1i 11 . . . . . 6  |-  ( ph  ->  0  e.  RR )
4744, 37, 46ltsubaddd 9554 . . . . 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 6028 . . . . . . 7  |-  ( q  =  0 p  -> 
( G  o F  x.  q )  =  ( G  o F  x.  0 p ) )
5352oveq2d 6036 . . . . . 6  |-  ( q  =  0 p  -> 
( F  o F  -  ( G  o F  x.  q )
)  =  ( F  o F  -  ( G  o F  x.  0 p ) ) )
5451, 53syl5eq 2431 . . . . 5  |-  ( q  =  0 p  ->  R  =  ( F  o F  -  ( G  o F  x.  0 p ) ) )
5554eqeq1d 2395 . . . 4  |-  ( q  =  0 p  -> 
( R  =  0 p  <->  ( F  o F  -  ( G  o F  x.  0 p ) )  =  0 p ) )
5654fveq2d 5672 . . . . 5  |-  ( q  =  0 p  -> 
(deg `  R )  =  (deg `  ( F  o F  -  ( G  o F  x.  0 p ) ) ) )
5756breq1d 4163 . . . 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 2995 . 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 1717    =/= wne 2550   E.wrex 2650   _Vcvv 2899    C_ wss 3263   class class class wbr 4153    Fn wfn 5389   -->wf 5390   ` cfv 5394  (class class class)co 6020    o Fcof 6242   CCcc 8921   RRcr 8922   0cc0 8923   1c1 8924    + caddc 8926    x. cmul 8928    < clt 9053    - cmin 9223   -ucneg 9224    / cdiv 9609   NN0cn0 10153   0 pc0p 19428  Polycply 19970  degcdgr 19973
This theorem is referenced by:  plydivex  20081
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-inf2 7529  ax-cnex 8979  ax-resscn 8980  ax-1cn 8981  ax-icn 8982  ax-addcl 8983  ax-addrcl 8984  ax-mulcl 8985  ax-mulrcl 8986  ax-mulcom 8987  ax-addass 8988  ax-mulass 8989  ax-distr 8990  ax-i2m1 8991  ax-1ne0 8992  ax-1rid 8993  ax-rnegex 8994  ax-rrecex 8995  ax-cnre 8996  ax-pre-lttri 8997  ax-pre-lttrn 8998  ax-pre-ltadd 8999  ax-pre-mulgt0 9000  ax-pre-sup 9001  ax-addf 9002
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-se 4483  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-isom 5403  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-of 6244  df-1st 6288  df-2nd 6289  df-riota 6485  df-recs 6569  df-rdg 6604  df-1o 6660  df-oadd 6664  df-er 6841  df-map 6956  df-pm 6957  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-sup 7381  df-oi 7412  df-card 7759  df-pnf 9055  df-mnf 9056  df-xr 9057  df-ltxr 9058  df-le 9059  df-sub 9225  df-neg 9226  df-div 9610  df-nn 9933  df-2 9990  df-3 9991  df-n0 10154  df-z 10215  df-uz 10421  df-rp 10545  df-fz 10976  df-fzo 11066  df-fl 11129  df-seq 11251  df-exp 11310  df-hash 11546  df-cj 11831  df-re 11832  df-im 11833  df-sqr 11967  df-abs 11968  df-clim 12209  df-rlim 12210  df-sum 12407  df-0p 19429  df-ply 19974  df-coe 19976  df-dgr 19977
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