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Theorem plydivlem3 21646
Description: Lemma for plydivex 21648. 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 21547 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
3 ply0 21561 . . 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 9351 . . . . . . 7  |-  CC  e.  _V
76a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
8 plyf 21551 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
9 ffn 5547 . . . . . . 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 21551 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
13 ffn 5547 . . . . . . . 8  |-  ( G : CC --> CC  ->  G  Fn  CC )
1411, 12, 133syl 20 . . . . . . 7  |-  ( ph  ->  G  Fn  CC )
15 plyf 21551 . . . . . . . 8  |-  ( 0p  e.  (Poly `  S )  ->  0p : CC --> CC )
16 ffn 5547 . . . . . . . 8  |-  ( 0p : CC --> CC  ->  0p  Fn  CC )
174, 15, 163syl 20 . . . . . . 7  |-  ( ph  ->  0p  Fn  CC )
18 inidm 3547 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
1914, 17, 7, 7, 18offn 6320 . . . . . 6  |-  ( ph  ->  ( G  oF  x.  0p )  Fn  CC )
20 eqidd 2434 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  =  ( F `  z
) )
21 eqidd 2434 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  =  ( G `  z
) )
22 0pval 20991 . . . . . . . . 9  |-  ( z  e.  CC  ->  (
0p `  z
)  =  0 )
2322adantl 463 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0p `  z )  =  0 )
2414, 17, 7, 7, 18, 21, 23ofval 6318 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  0p ) `  z )  =  ( ( G `  z
)  x.  0 ) )
2511, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  G : CC --> CC )
2625ffvelrnda 5831 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
2726mul01d 9556 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z )  x.  0 )  =  0 )
2824, 27eqtrd 2465 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  0p ) `  z )  =  0 )
291, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
3029ffvelrnda 5831 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
3130subid1d 9696 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  0 )  =  ( F `  z
) )
327, 10, 19, 10, 20, 28, 31offveq 6330 . . . . 5  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  0p
) )  =  F )
3332eqeq1d 2441 . . . 4  |-  ( ph  ->  ( ( F  oF  -  ( G  oF  x.  0p ) )  =  0p  <->  F  = 
0p ) )
3432fveq2d 5683 . . . . . 6  |-  ( ph  ->  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  =  (deg `  F
) )
35 dgrcl 21586 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3611, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3736nn0red 10625 . . . . . . . . 9  |-  ( ph  ->  (deg `  G )  e.  RR )
3837recnd 9400 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  CC )
3938addid2d 9558 . . . . . . 7  |-  ( ph  ->  ( 0  +  (deg
`  G ) )  =  (deg `  G
) )
4039eqcomd 2438 . . . . . 6  |-  ( ph  ->  (deg `  G )  =  ( 0  +  (deg `  G )
) )
4134, 40breq12d 4293 . . . . 5  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  <  (deg `  G
)  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
42 dgrcl 21586 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
431, 42syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  F )  e.  NN0 )
4443nn0red 10625 . . . . . 6  |-  ( ph  ->  (deg `  F )  e.  RR )
45 0red 9375 . . . . . 6  |-  ( ph  ->  0  e.  RR )
4644, 37, 45ltsubaddd 9923 . . . . 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 702 . . 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 6088 . . . . . . 7  |-  ( q  =  0p  -> 
( G  oF  x.  q )  =  ( G  oF  x.  0p ) )
5251oveq2d 6096 . . . . . 6  |-  ( q  =  0p  -> 
( F  oF  -  ( G  oF  x.  q )
)  =  ( F  oF  -  ( G  oF  x.  0p ) ) )
5350, 52syl5eq 2477 . . . . 5  |-  ( q  =  0p  ->  R  =  ( F  oF  -  ( G  oF  x.  0p ) ) )
5453eqeq1d 2441 . . . 4  |-  ( q  =  0p  -> 
( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  0p ) )  =  0p ) )
5553fveq2d 5683 . . . . 5  |-  ( q  =  0p  -> 
(deg `  R )  =  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) ) )
5655breq1d 4290 . . . 4  |-  ( q  =  0p  -> 
( (deg `  R
)  <  (deg `  G
)  <->  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  <  (deg `  G
) ) )
5754, 56orbi12d 702 . . 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 3062 . 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 654 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 1362    e. wcel 1755    =/= wne 2596   E.wrex 2706   _Vcvv 2962    C_ wss 3316   class class class wbr 4280    Fn wfn 5401   -->wf 5402   ` cfv 5406  (class class class)co 6080    oFcof 6307   CCcc 9268   0cc0 9270   1c1 9271    + caddc 9273    x. cmul 9275    < clt 9406    - cmin 9583   -ucneg 9584    / cdiv 9981   NN0cn0 10567   0pc0p 20989  Polycply 21537  degcdgr 21540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347  ax-pre-sup 9348  ax-addf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-fal 1368  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-om 6466  df-1st 6566  df-2nd 6567  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-sup 7679  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-div 9982  df-nn 10311  df-2 10368  df-3 10369  df-n0 10568  df-z 10635  df-uz 10850  df-rp 10980  df-fz 11425  df-fzo 11533  df-fl 11626  df-seq 11791  df-exp 11850  df-hash 12088  df-cj 12572  df-re 12573  df-im 12574  df-sqr 12708  df-abs 12709  df-clim 12950  df-rlim 12951  df-sum 13148  df-0p 20990  df-ply 21541  df-coe 21543  df-dgr 21544
This theorem is referenced by:  plydivex  21648
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