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Theorem plydivlem3 22535
Description: Lemma for plydivex 22537. 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  =/=  0p )
plydiv.r  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
plydiv.0  |-  ( ph  ->  ( F  =  0p  \/  ( (deg
`  F )  -  (deg `  G ) )  <  0 ) )
Assertion
Ref Expression
plydivlem3  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (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 22436 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
3 ply0 22450 . . 3  |-  ( S 
C_  CC  ->  0p  e.  (Poly `  S
) )
41, 2, 33syl 20 . 2  |-  ( ph  ->  0p  e.  (Poly `  S ) )
5 plydiv.0 . . 3  |-  ( ph  ->  ( F  =  0p  \/  ( (deg
`  F )  -  (deg `  G ) )  <  0 ) )
6 cnex 9583 . . . . . . 7  |-  CC  e.  _V
76a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
8 plyf 22440 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
9 ffn 5736 . . . . . . 7  |-  ( F : CC --> CC  ->  F  Fn  CC )
101, 8, 93syl 20 . . . . . 6  |-  ( ph  ->  F  Fn  CC )
11 plydiv.g . . . . . . . 8  |-  ( ph  ->  G  e.  (Poly `  S ) )
12 plyf 22440 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
13 ffn 5736 . . . . . . . 8  |-  ( G : CC --> CC  ->  G  Fn  CC )
1411, 12, 133syl 20 . . . . . . 7  |-  ( ph  ->  G  Fn  CC )
15 plyf 22440 . . . . . . . 8  |-  ( 0p  e.  (Poly `  S )  ->  0p : CC --> CC )
16 ffn 5736 . . . . . . . 8  |-  ( 0p : CC --> CC  ->  0p  Fn  CC )
174, 15, 163syl 20 . . . . . . 7  |-  ( ph  ->  0p  Fn  CC )
18 inidm 3712 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
1914, 17, 7, 7, 18offn 6545 . . . . . 6  |-  ( ph  ->  ( G  oF  x.  0p )  Fn  CC )
20 eqidd 2468 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  =  ( F `  z
) )
21 eqidd 2468 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  =  ( G `  z
) )
22 0pval 21923 . . . . . . . . 9  |-  ( z  e.  CC  ->  (
0p `  z
)  =  0 )
2322adantl 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0p `  z )  =  0 )
2414, 17, 7, 7, 18, 21, 23ofval 6543 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  0p ) `  z )  =  ( ( G `  z
)  x.  0 ) )
2511, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  G : CC --> CC )
2625ffvelrnda 6031 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
2726mul01d 9788 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z )  x.  0 )  =  0 )
2824, 27eqtrd 2508 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  0p ) `  z )  =  0 )
291, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
3029ffvelrnda 6031 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
3130subid1d 9929 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  0 )  =  ( F `  z
) )
327, 10, 19, 10, 20, 28, 31offveq 6555 . . . . 5  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  0p
) )  =  F )
3332eqeq1d 2469 . . . 4  |-  ( ph  ->  ( ( F  oF  -  ( G  oF  x.  0p ) )  =  0p  <->  F  = 
0p ) )
3432fveq2d 5875 . . . . . 6  |-  ( ph  ->  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  =  (deg `  F
) )
35 dgrcl 22475 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3611, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3736nn0red 10863 . . . . . . . . 9  |-  ( ph  ->  (deg `  G )  e.  RR )
3837recnd 9632 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  CC )
3938addid2d 9790 . . . . . . 7  |-  ( ph  ->  ( 0  +  (deg
`  G ) )  =  (deg `  G
) )
4039eqcomd 2475 . . . . . 6  |-  ( ph  ->  (deg `  G )  =  ( 0  +  (deg `  G )
) )
4134, 40breq12d 4465 . . . . 5  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  <  (deg `  G
)  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
42 dgrcl 22475 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
431, 42syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  F )  e.  NN0 )
4443nn0red 10863 . . . . . 6  |-  ( ph  ->  (deg `  F )  e.  RR )
45 0red 9607 . . . . . 6  |-  ( ph  ->  0  e.  RR )
4644, 37, 45ltsubaddd 10158 . . . . 5  |-  ( ph  ->  ( ( (deg `  F )  -  (deg `  G ) )  <  0  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
4741, 46bitr4d 256 . . . 4  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  <  (deg `  G
)  <->  ( (deg `  F )  -  (deg `  G ) )  <  0 ) )
4833, 47orbi12d 709 . . 3  |-  ( ph  ->  ( ( ( F  oF  -  ( G  oF  x.  0p ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  0p
) ) )  < 
(deg `  G )
)  <->  ( F  =  0p  \/  (
(deg `  F )  -  (deg `  G )
)  <  0 ) ) )
495, 48mpbird 232 . 2  |-  ( ph  ->  ( ( F  oF  -  ( G  oF  x.  0p ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  0p
) ) )  < 
(deg `  G )
) )
50 plydiv.r . . . . . 6  |-  R  =  ( F  oF  -  ( G  oF  x.  q )
)
51 oveq2 6302 . . . . . . 7  |-  ( q  =  0p  -> 
( G  oF  x.  q )  =  ( G  oF  x.  0p ) )
5251oveq2d 6310 . . . . . 6  |-  ( q  =  0p  -> 
( F  oF  -  ( G  oF  x.  q )
)  =  ( F  oF  -  ( G  oF  x.  0p ) ) )
5350, 52syl5eq 2520 . . . . 5  |-  ( q  =  0p  ->  R  =  ( F  oF  -  ( G  oF  x.  0p ) ) )
5453eqeq1d 2469 . . . 4  |-  ( q  =  0p  -> 
( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  0p ) )  =  0p ) )
5553fveq2d 5875 . . . . 5  |-  ( q  =  0p  -> 
(deg `  R )  =  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) ) )
5655breq1d 4462 . . . 4  |-  ( q  =  0p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  <  (deg `  G
) ) )
5754, 56orbi12d 709 . . 3  |-  ( q  =  0p  -> 
( ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) )  <->  ( ( F  oF  -  ( G  oF  x.  0p ) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  0p
) ) )  < 
(deg `  G )
) ) )
5857rspcev 3219 . 2  |-  ( ( 0p  e.  (Poly `  S )  /\  (
( F  oF  -  ( G  oF  x.  0p
) )  =  0p  \/  (deg `  ( F  oF  -  ( G  oF  x.  0p
) ) )  < 
(deg `  G )
) )  ->  E. q  e.  (Poly `  S )
( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) )
594, 49, 58syl2anc 661 1  |-  ( ph  ->  E. q  e.  (Poly `  S ) ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2818   _Vcvv 3118    C_ wss 3481   class class class wbr 4452    Fn wfn 5588   -->wf 5589   ` cfv 5593  (class class class)co 6294    oFcof 6532   CCcc 9500   0cc0 9502   1c1 9503    + caddc 9505    x. cmul 9507    < clt 9638    - cmin 9815   -ucneg 9816    / cdiv 10216   NN0cn0 10805   0pc0p 21921  Polycply 22426  degcdgr 22429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586  ax-inf2 8068  ax-cnex 9558  ax-resscn 9559  ax-1cn 9560  ax-icn 9561  ax-addcl 9562  ax-addrcl 9563  ax-mulcl 9564  ax-mulrcl 9565  ax-mulcom 9566  ax-addass 9567  ax-mulass 9568  ax-distr 9569  ax-i2m1 9570  ax-1ne0 9571  ax-1rid 9572  ax-rnegex 9573  ax-rrecex 9574  ax-cnre 9575  ax-pre-lttri 9576  ax-pre-lttrn 9577  ax-pre-ltadd 9578  ax-pre-mulgt0 9579  ax-pre-sup 9580  ax-addf 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4251  df-int 4288  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-se 4844  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-isom 5602  df-riota 6255  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-of 6534  df-om 6695  df-1st 6794  df-2nd 6795  df-recs 7052  df-rdg 7086  df-1o 7140  df-oadd 7144  df-er 7321  df-map 7432  df-pm 7433  df-en 7527  df-dom 7528  df-sdom 7529  df-fin 7530  df-sup 7911  df-oi 7945  df-card 8330  df-pnf 9640  df-mnf 9641  df-xr 9642  df-ltxr 9643  df-le 9644  df-sub 9817  df-neg 9818  df-div 10217  df-nn 10547  df-2 10604  df-3 10605  df-n0 10806  df-z 10875  df-uz 11093  df-rp 11231  df-fz 11683  df-fzo 11803  df-fl 11907  df-seq 12086  df-exp 12145  df-hash 12384  df-cj 12907  df-re 12908  df-im 12909  df-sqrt 13043  df-abs 13044  df-clim 13286  df-rlim 13287  df-sum 13484  df-0p 21922  df-ply 22430  df-coe 22432  df-dgr 22433
This theorem is referenced by:  plydivex  22537
  Copyright terms: Public domain W3C validator