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Theorem plydivlem3 22556
Description: Lemma for plydivex 22558. 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 22457 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
3 ply0 22471 . . 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 9571 . . . . . . 7  |-  CC  e.  _V
76a1i 11 . . . . . 6  |-  ( ph  ->  CC  e.  _V )
8 plyf 22461 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  F : CC
--> CC )
9 ffn 5717 . . . . . . 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 22461 . . . . . . . 8  |-  ( G  e.  (Poly `  S
)  ->  G : CC
--> CC )
13 ffn 5717 . . . . . . . 8  |-  ( G : CC --> CC  ->  G  Fn  CC )
1411, 12, 133syl 20 . . . . . . 7  |-  ( ph  ->  G  Fn  CC )
15 plyf 22461 . . . . . . . 8  |-  ( 0p  e.  (Poly `  S )  ->  0p : CC --> CC )
16 ffn 5717 . . . . . . . 8  |-  ( 0p : CC --> CC  ->  0p  Fn  CC )
174, 15, 163syl 20 . . . . . . 7  |-  ( ph  ->  0p  Fn  CC )
18 inidm 3689 . . . . . . 7  |-  ( CC 
i^i  CC )  =  CC
1914, 17, 7, 7, 18offn 6532 . . . . . 6  |-  ( ph  ->  ( G  oF  x.  0p )  Fn  CC )
20 eqidd 2442 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  =  ( F `  z
) )
21 eqidd 2442 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  =  ( G `  z
) )
22 0pval 21944 . . . . . . . . 9  |-  ( z  e.  CC  ->  (
0p `  z
)  =  0 )
2322adantl 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( 0p `  z )  =  0 )
2414, 17, 7, 7, 18, 21, 23ofval 6530 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  0p ) `  z )  =  ( ( G `  z
)  x.  0 ) )
2511, 12syl 16 . . . . . . . . 9  |-  ( ph  ->  G : CC --> CC )
2625ffvelrnda 6012 . . . . . . . 8  |-  ( (
ph  /\  z  e.  CC )  ->  ( G `
 z )  e.  CC )
2726mul01d 9777 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G `  z )  x.  0 )  =  0 )
2824, 27eqtrd 2482 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( G  oF  x.  0p ) `  z )  =  0 )
291, 8syl 16 . . . . . . . 8  |-  ( ph  ->  F : CC --> CC )
3029ffvelrnda 6012 . . . . . . 7  |-  ( (
ph  /\  z  e.  CC )  ->  ( F `
 z )  e.  CC )
3130subid1d 9920 . . . . . 6  |-  ( (
ph  /\  z  e.  CC )  ->  ( ( F `  z )  -  0 )  =  ( F `  z
) )
327, 10, 19, 10, 20, 28, 31offveq 6542 . . . . 5  |-  ( ph  ->  ( F  oF  -  ( G  oF  x.  0p
) )  =  F )
3332eqeq1d 2443 . . . 4  |-  ( ph  ->  ( ( F  oF  -  ( G  oF  x.  0p ) )  =  0p  <->  F  = 
0p ) )
3432fveq2d 5856 . . . . . 6  |-  ( ph  ->  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  =  (deg `  F
) )
35 dgrcl 22496 . . . . . . . . . . 11  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
3611, 35syl 16 . . . . . . . . . 10  |-  ( ph  ->  (deg `  G )  e.  NN0 )
3736nn0red 10854 . . . . . . . . 9  |-  ( ph  ->  (deg `  G )  e.  RR )
3837recnd 9620 . . . . . . . 8  |-  ( ph  ->  (deg `  G )  e.  CC )
3938addid2d 9779 . . . . . . 7  |-  ( ph  ->  ( 0  +  (deg
`  G ) )  =  (deg `  G
) )
4039eqcomd 2449 . . . . . 6  |-  ( ph  ->  (deg `  G )  =  ( 0  +  (deg `  G )
) )
4134, 40breq12d 4446 . . . . 5  |-  ( ph  ->  ( (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) )  <  (deg `  G
)  <->  (deg `  F )  <  ( 0  +  (deg
`  G ) ) ) )
42 dgrcl 22496 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
431, 42syl 16 . . . . . . 7  |-  ( ph  ->  (deg `  F )  e.  NN0 )
4443nn0red 10854 . . . . . 6  |-  ( ph  ->  (deg `  F )  e.  RR )
45 0red 9595 . . . . . 6  |-  ( ph  ->  0  e.  RR )
4644, 37, 45ltsubaddd 10149 . . . . 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 6285 . . . . . . 7  |-  ( q  =  0p  -> 
( G  oF  x.  q )  =  ( G  oF  x.  0p ) )
5251oveq2d 6293 . . . . . 6  |-  ( q  =  0p  -> 
( F  oF  -  ( G  oF  x.  q )
)  =  ( F  oF  -  ( G  oF  x.  0p ) ) )
5350, 52syl5eq 2494 . . . . 5  |-  ( q  =  0p  ->  R  =  ( F  oF  -  ( G  oF  x.  0p ) ) )
5453eqeq1d 2443 . . . 4  |-  ( q  =  0p  -> 
( R  =  0p  <->  ( F  oF  -  ( G  oF  x.  0p ) )  =  0p ) )
5553fveq2d 5856 . . . . 5  |-  ( q  =  0p  -> 
(deg `  R )  =  (deg `  ( F  oF  -  ( G  oF  x.  0p ) ) ) )
5655breq1d 4443 . . . 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 3194 . 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 1381    e. wcel 1802    =/= wne 2636   E.wrex 2792   _Vcvv 3093    C_ wss 3458   class class class wbr 4433    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    oFcof 6519   CCcc 9488   0cc0 9490   1c1 9491    + caddc 9493    x. cmul 9495    < clt 9626    - cmin 9805   -ucneg 9806    / cdiv 10207   NN0cn0 10796   0pc0p 21942  Polycply 22447  degcdgr 22450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568  ax-addf 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-fal 1387  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-se 4825  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-isom 5583  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-map 7420  df-pm 7421  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-sup 7899  df-oi 7933  df-card 8318  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-3 10596  df-n0 10797  df-z 10866  df-uz 11086  df-rp 11225  df-fz 11677  df-fzo 11799  df-fl 11903  df-seq 12082  df-exp 12141  df-hash 12380  df-cj 12906  df-re 12907  df-im 12908  df-sqrt 13042  df-abs 13043  df-clim 13285  df-rlim 13286  df-sum 13483  df-0p 21943  df-ply 22451  df-coe 22453  df-dgr 22454
This theorem is referenced by:  plydivex  22558
  Copyright terms: Public domain W3C validator