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Theorem dgrlt 20176
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 5724 . . . . . 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 20172 . . . . . . 7  |-  (deg ` 
0 p )  =  0
54eqcomi 2439 . . . . . 6  |-  0  =  (deg `  0 p
)
62, 3, 53eqtr4g 2492 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  N  =  0 )
7 nn0ge0 10239 . . . . . 6  |-  ( M  e.  NN0  ->  0  <_  M )
87ad2antlr 708 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  0  <_  M
)
96, 8eqbrtrd 4224 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  N  <_  M
)
101fveq2d 5724 . . . . . . 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 20166 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
1312eqcomi 2439 . . . . . . 7  |-  ( NN0 
X.  { 0 } )  =  (coeff ` 
0 p )
1410, 11, 133eqtr4g 2492 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  A  =  ( NN0  X.  { 0 } ) )
1514fveq1d 5722 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( A `  M )  =  ( ( NN0  X.  {
0 } ) `  M ) )
16 c0ex 9077 . . . . . . 7  |-  0  e.  _V
1716fvconst2 5939 . . . . . 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 2467 . . . 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 20144 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
223, 21syl5eqel 2519 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
2322nn0red 10267 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  N  e.  RR )
24 nn0re 10222 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  RR )
25 ltle 9155 . . . . . 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 20145 . . . . . . . 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 9146 . . . . . . . 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 2659 . . . . 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 9153 . . . . . . 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 5720 . . . . . 6  |-  ( N  =  M  ->  ( A `  N )  =  ( A `  M ) )
423, 11dgreq0 20175 . . . . . . . 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 2446 . . . . . . 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 1652    e. wcel 1725    =/= wne 2598   {csn 3806   class class class wbr 4204    X. cxp 4868   ` cfv 5446   RRcr 8981   0cc0 8982    < clt 9112    <_ cle 9113   NN0cn0 10213   0 pc0p 19553  Polycply 20095  coeffccoe 20097  degcdgr 20098
This theorem is referenced by:  dgrcolem2  20184  plydivlem4  20205  plydiveu  20207  dgrsub2  27307
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-hash 11611  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-clim 12274  df-rlim 12275  df-sum 12472  df-0p 19554  df-ply 20099  df-coe 20101  df-dgr 20102
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