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Theorem quotdgr 21912
Description: Remainder property of the quotient function. (Contributed by Mario Carneiro, 26-Jul-2014.)
Hypothesis
Ref Expression
quotdgr.1  |-  R  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
Assertion
Ref Expression
quotdgr  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )

Proof of Theorem quotdgr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addcl 9479 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  +  y )  e.  CC )
21adantl 466 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  +  y )  e.  CC )
3 mulcl 9481 . . . 4  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
43adantl 466 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  /\  (
x  e.  CC  /\  y  e.  CC )
)  ->  ( x  x.  y )  e.  CC )
5 reccl 10116 . . . 4  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  CC )
65adantl 466 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  /\  (
x  e.  CC  /\  x  =/=  0 ) )  ->  ( 1  /  x )  e.  CC )
7 neg1cn 10540 . . . 4  |-  -u 1  e.  CC
87a1i 11 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  -u 1  e.  CC )
9 plyssc 21811 . . . 4  |-  (Poly `  S )  C_  (Poly `  CC )
10 simp1 988 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  S )
)
119, 10sseldi 3465 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  F  e.  (Poly `  CC )
)
12 simp2 989 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  S )
)
139, 12sseldi 3465 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  e.  (Poly `  CC )
)
14 simp3 990 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  G  =/=  0p )
15 quotdgr.1 . . 3  |-  R  =  ( F  oF  -  ( G  oF  x.  ( F quot  G ) ) )
162, 4, 6, 8, 11, 13, 14, 15quotlem 21909 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  (
( F quot  G )  e.  (Poly `  CC )  /\  ( R  =  0p  \/  (deg `  R )  <  (deg `  G ) ) ) )
1716simprd 463 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  G  =/=  0p )  ->  ( R  =  0p  \/  (deg `  R )  <  (deg `  G )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4403   ` cfv 5529  (class class class)co 6203    oFcof 6431   CCcc 9395   0cc0 9397   1c1 9398    + caddc 9400    x. cmul 9402    < clt 9533    - cmin 9710   -ucneg 9711    / cdiv 10108   0pc0p 21290  Polycply 21795  degcdgr 21798   quot cquot 21899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-rlim 13089  df-sum 13286  df-0p 21291  df-ply 21799  df-coe 21801  df-dgr 21802  df-quot 21900
This theorem is referenced by:  plyrem  21914  quotcan  21918
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