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Theorem dgrlt 20044
Description: Two ways to say that the degree of  F is strictly less than  N. (Contributed by Mario Carneiro, 25-Jul-2014.)
Hypotheses
Ref Expression
dgreq0.1  |-  N  =  (deg `  F )
dgreq0.2  |-  A  =  (coeff `  F )
Assertion
Ref Expression
dgrlt  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( F  =  0 p  \/  N  < 
M )  <->  ( N  <_  M  /\  ( A `
 M )  =  0 ) ) )

Proof of Theorem dgrlt
StepHypRef Expression
1 simpr 448 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  F  =  0 p )
21fveq2d 5665 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  (deg `  F
)  =  (deg ` 
0 p ) )
3 dgreq0.1 . . . . . 6  |-  N  =  (deg `  F )
4 dgr0 20040 . . . . . . 7  |-  (deg ` 
0 p )  =  0
54eqcomi 2384 . . . . . 6  |-  0  =  (deg `  0 p
)
62, 3, 53eqtr4g 2437 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  N  =  0 )
7 nn0ge0 10172 . . . . . 6  |-  ( M  e.  NN0  ->  0  <_  M )
87ad2antlr 708 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  0  <_  M
)
96, 8eqbrtrd 4166 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  N  <_  M
)
101fveq2d 5665 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  (coeff `  F
)  =  (coeff ` 
0 p ) )
11 dgreq0.2 . . . . . . 7  |-  A  =  (coeff `  F )
12 coe0 20034 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
1312eqcomi 2384 . . . . . . 7  |-  ( NN0 
X.  { 0 } )  =  (coeff ` 
0 p )
1410, 11, 133eqtr4g 2437 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  A  =  ( NN0  X.  { 0 } ) )
1514fveq1d 5663 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( A `  M )  =  ( ( NN0  X.  {
0 } ) `  M ) )
16 c0ex 9011 . . . . . . 7  |-  0  e.  _V
1716fvconst2 5879 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  M
)  =  0 )
1817ad2antlr 708 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( ( NN0 
X.  { 0 } ) `  M )  =  0 )
1915, 18eqtrd 2412 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( A `  M )  =  0 )
209, 19jca 519 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
21 dgrcl 20012 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
223, 21syl5eqel 2464 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
2322nn0red 10200 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  N  e.  RR )
24 nn0re 10155 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  RR )
25 ltle 9089 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <  M  ->  N  <_  M )
)
2623, 24, 25syl2an 464 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  N  <_  M ) )
2726imp 419 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  N  <_  M )
2811, 3dgrub 20013 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )
29283expia 1155 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  M  <_  N ) )
30 lenlt 9080 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3124, 23, 30syl2anr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3229, 31sylibd 206 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  -.  N  <  M ) )
3332necon4ad 2604 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  ( A `  M )  =  0 ) )
3433imp 419 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( A `  M )  =  0 )
3527, 34jca 519 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
3620, 35jaodan 761 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( F  =  0 p  \/  N  <  M ) )  ->  ( N  <_  M  /\  ( A `
 M )  =  0 ) )
37 leloe 9087 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M )
) )
3823, 24, 37syl2an 464 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M ) ) )
3938biimpa 471 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <_  M )  ->  ( N  <  M  \/  N  =  M ) )
4039adantrr 698 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  N  =  M ) )
41 fveq2 5661 . . . . . 6  |-  ( N  =  M  ->  ( A `  N )  =  ( A `  M ) )
423, 11dgreq0 20043 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
4342ad2antrr 707 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
44 simprr 734 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( A `  M )  =  0 )
4544eqeq2d 2391 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( A `
 N )  =  ( A `  M
)  <->  ( A `  N )  =  0 ) )
4643, 45bitr4d 248 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0 p  <->  ( A `  N )  =  ( A `  M ) ) )
4741, 46syl5ibr 213 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  =  M  ->  F  = 
0 p ) )
4847orim2d 814 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( N  <  M  \/  N  =  M )  ->  ( N  <  M  \/  F  =  0 p ) ) )
4940, 48mpd 15 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  F  =  0 p ) )
5049orcomd 378 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0 p  \/  N  <  M ) )
5136, 50impbida 806 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( F  =  0 p  \/  N  < 
M )  <->  ( N  <_  M  /\  ( A `
 M )  =  0 ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2543   {csn 3750   class class class wbr 4146    X. cxp 4809   ` cfv 5387   RRcr 8915   0cc0 8916    < clt 9046    <_ cle 9047   NN0cn0 10146   0 pc0p 19421  Polycply 19963  coeffccoe 19965  degcdgr 19966
This theorem is referenced by:  dgrcolem2  20052  plydivlem4  20073  plydiveu  20075  dgrsub2  27001
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-inf2 7522  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994  ax-addf 8995
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-int 3986  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-se 4476  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-isom 5396  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-of 6237  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-oadd 6657  df-er 6834  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-sup 7374  df-oi 7405  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-div 9603  df-nn 9926  df-2 9983  df-3 9984  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-fl 11122  df-seq 11244  df-exp 11303  df-hash 11539  df-cj 11824  df-re 11825  df-im 11826  df-sqr 11960  df-abs 11961  df-clim 12202  df-rlim 12203  df-sum 12400  df-0p 19422  df-ply 19967  df-coe 19969  df-dgr 19970
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