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Theorem dgrlt 23088
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  =  0p  \/  N  < 
M )  <->  ( N  <_  M  /\  ( A `
 M )  =  0 ) ) )

Proof of Theorem dgrlt
StepHypRef Expression
1 simpr 462 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  F  =  0p )
21fveq2d 5885 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  (deg `  F
)  =  (deg ` 
0p ) )
3 dgreq0.1 . . . . . 6  |-  N  =  (deg `  F )
4 dgr0 23084 . . . . . . 7  |-  (deg ` 
0p )  =  0
54eqcomi 2442 . . . . . 6  |-  0  =  (deg `  0p
)
62, 3, 53eqtr4g 2495 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  N  =  0 )
7 nn0ge0 10895 . . . . . 6  |-  ( M  e.  NN0  ->  0  <_  M )
87ad2antlr 731 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  0  <_  M
)
96, 8eqbrtrd 4446 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  N  <_  M
)
101fveq2d 5885 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  (coeff `  F
)  =  (coeff ` 
0p ) )
11 dgreq0.2 . . . . . . 7  |-  A  =  (coeff `  F )
12 coe0 23078 . . . . . . . 8  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
1312eqcomi 2442 . . . . . . 7  |-  ( NN0 
X.  { 0 } )  =  (coeff ` 
0p )
1410, 11, 133eqtr4g 2495 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  A  =  ( NN0  X.  { 0 } ) )
1514fveq1d 5883 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( A `  M )  =  ( ( NN0  X.  {
0 } ) `  M ) )
16 c0ex 9636 . . . . . . 7  |-  0  e.  _V
1716fvconst2 6135 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  M
)  =  0 )
1817ad2antlr 731 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( ( NN0 
X.  { 0 } ) `  M )  =  0 )
1915, 18eqtrd 2470 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( A `  M )  =  0 )
209, 19jca 534 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
21 dgrcl 23055 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
223, 21syl5eqel 2521 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
2322nn0red 10926 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  N  e.  RR )
24 nn0re 10878 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  RR )
25 ltle 9721 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <  M  ->  N  <_  M )
)
2623, 24, 25syl2an 479 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  N  <_  M ) )
2726imp 430 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  N  <_  M )
2811, 3dgrub 23056 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )
29283expia 1207 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  M  <_  N ) )
30 lenlt 9711 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3124, 23, 30syl2anr 480 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3229, 31sylibd 217 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  -.  N  <  M ) )
3332necon4ad 2651 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  ( A `  M )  =  0 ) )
3433imp 430 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( A `  M )  =  0 )
3527, 34jca 534 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
3620, 35jaodan 792 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( F  =  0p  \/  N  <  M ) )  ->  ( N  <_  M  /\  ( A `
 M )  =  0 ) )
37 leloe 9719 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M )
) )
3823, 24, 37syl2an 479 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M ) ) )
3938biimpa 486 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <_  M )  ->  ( N  <  M  \/  N  =  M ) )
4039adantrr 721 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  N  =  M ) )
41 fveq2 5881 . . . . . 6  |-  ( N  =  M  ->  ( A `  N )  =  ( A `  M ) )
423, 11dgreq0 23087 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
4342ad2antrr 730 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
44 simprr 764 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( A `  M )  =  0 )
4544eqeq2d 2443 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( A `
 N )  =  ( A `  M
)  <->  ( A `  N )  =  0 ) )
4643, 45bitr4d 259 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0p  <->  ( A `  N )  =  ( A `  M ) ) )
4741, 46syl5ibr 224 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  =  M  ->  F  = 
0p ) )
4847orim2d 848 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( N  <  M  \/  N  =  M )  ->  ( N  <  M  \/  F  =  0p ) ) )
4940, 48mpd 15 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  F  =  0p ) )
5049orcomd 389 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0p  \/  N  <  M ) )
5136, 50impbida 840 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( F  =  0p  \/  N  < 
M )  <->  ( N  <_  M  /\  ( A `
 M )  =  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   {csn 4002   class class class wbr 4426    X. cxp 4852   ` cfv 5601   RRcr 9537   0cc0 9538    < clt 9674    <_ cle 9675   NN0cn0 10869   0pc0p 22504  Polycply 23006  coeffccoe 23008  degcdgr 23009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-clim 13530  df-rlim 13531  df-sum 13731  df-0p 22505  df-ply 23010  df-coe 23012  df-dgr 23013
This theorem is referenced by:  dgrcolem2  23096  plydivlem4  23117  plydiveu  23119  dgrsub2  35700  elaa2lem  37665
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