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Theorem dgrlt 19647
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 447 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  F  =  0 p )
21fveq2d 5529 . . . . . 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 19643 . . . . . . 7  |-  (deg ` 
0 p )  =  0
54eqcomi 2287 . . . . . 6  |-  0  =  (deg `  0 p
)
62, 3, 53eqtr4g 2340 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  N  =  0 )
7 nn0ge0 9991 . . . . . 6  |-  ( M  e.  NN0  ->  0  <_  M )
87ad2antlr 707 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  0  <_  M
)
96, 8eqbrtrd 4043 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  N  <_  M
)
101fveq2d 5529 . . . . . . 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 19637 . . . . . . . 8  |-  (coeff ` 
0 p )  =  ( NN0  X.  {
0 } )
1312eqcomi 2287 . . . . . . 7  |-  ( NN0 
X.  { 0 } )  =  (coeff ` 
0 p )
1410, 11, 133eqtr4g 2340 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  A  =  ( NN0  X.  { 0 } ) )
1514fveq1d 5527 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( A `  M )  =  ( ( NN0  X.  {
0 } ) `  M ) )
16 c0ex 8832 . . . . . . 7  |-  0  e.  _V
1716fvconst2 5729 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  M
)  =  0 )
1817ad2antlr 707 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( ( NN0 
X.  { 0 } ) `  M )  =  0 )
1915, 18eqtrd 2315 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( A `  M )  =  0 )
209, 19jca 518 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0 p )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
21 dgrcl 19615 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
223, 21syl5eqel 2367 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
2322nn0red 10019 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  N  e.  RR )
24 nn0re 9974 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  RR )
25 ltle 8910 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <  M  ->  N  <_  M )
)
2623, 24, 25syl2an 463 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  N  <_  M ) )
2726imp 418 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  N  <_  M )
2811, 3dgrub 19616 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )
29283expia 1153 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  M  <_  N ) )
30 lenlt 8901 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3124, 23, 30syl2anr 464 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3229, 31sylibd 205 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  -.  N  <  M ) )
3332necon4ad 2507 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  ( A `  M )  =  0 ) )
3433imp 418 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( A `  M )  =  0 )
3527, 34jca 518 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
3620, 35jaodan 760 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( F  =  0 p  \/  N  <  M ) )  ->  ( N  <_  M  /\  ( A `
 M )  =  0 ) )
37 leloe 8908 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M )
) )
3823, 24, 37syl2an 463 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M ) ) )
3938biimpa 470 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <_  M )  ->  ( N  <  M  \/  N  =  M ) )
4039adantrr 697 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  N  =  M ) )
41 fveq2 5525 . . . . . 6  |-  ( N  =  M  ->  ( A `  N )  =  ( A `  M ) )
423, 11dgreq0 19646 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
4342ad2antrr 706 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0 p  <->  ( A `  N )  =  0 ) )
44 simprr 733 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( A `  M )  =  0 )
4544eqeq2d 2294 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( A `
 N )  =  ( A `  M
)  <->  ( A `  N )  =  0 ) )
4643, 45bitr4d 247 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0 p  <->  ( A `  N )  =  ( A `  M ) ) )
4741, 46syl5ibr 212 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  =  M  ->  F  = 
0 p ) )
4847orim2d 813 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( N  <  M  \/  N  =  M )  ->  ( N  <  M  \/  F  =  0 p ) ) )
4940, 48mpd 14 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  F  =  0 p ) )
5049orcomd 377 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0 p  \/  N  <  M ) )
5136, 50impbida 805 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 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   {csn 3640   class class class wbr 4023    X. cxp 4687   ` cfv 5255   RRcr 8736   0cc0 8737    < clt 8867    <_ cle 8868   NN0cn0 9965   0 pc0p 19024  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dgrcolem2  19655  plydivlem4  19676  plydiveu  19678  dgrsub2  27339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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