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Theorem dgrlt 22425
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 461 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  F  =  0p )
21fveq2d 5870 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  (deg `  F
)  =  (deg ` 
0p ) )
3 dgreq0.1 . . . . . 6  |-  N  =  (deg `  F )
4 dgr0 22421 . . . . . . 7  |-  (deg ` 
0p )  =  0
54eqcomi 2480 . . . . . 6  |-  0  =  (deg `  0p
)
62, 3, 53eqtr4g 2533 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  N  =  0 )
7 nn0ge0 10821 . . . . . 6  |-  ( M  e.  NN0  ->  0  <_  M )
87ad2antlr 726 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  0  <_  M
)
96, 8eqbrtrd 4467 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  N  <_  M
)
101fveq2d 5870 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  (coeff `  F
)  =  (coeff ` 
0p ) )
11 dgreq0.2 . . . . . . 7  |-  A  =  (coeff `  F )
12 coe0 22415 . . . . . . . 8  |-  (coeff ` 
0p )  =  ( NN0  X.  {
0 } )
1312eqcomi 2480 . . . . . . 7  |-  ( NN0 
X.  { 0 } )  =  (coeff ` 
0p )
1410, 11, 133eqtr4g 2533 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  A  =  ( NN0  X.  { 0 } ) )
1514fveq1d 5868 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( A `  M )  =  ( ( NN0  X.  {
0 } ) `  M ) )
16 c0ex 9590 . . . . . . 7  |-  0  e.  _V
1716fvconst2 6116 . . . . . 6  |-  ( M  e.  NN0  ->  ( ( NN0  X.  { 0 } ) `  M
)  =  0 )
1817ad2antlr 726 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( ( NN0 
X.  { 0 } ) `  M )  =  0 )
1915, 18eqtrd 2508 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( A `  M )  =  0 )
209, 19jca 532 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  F  =  0p )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
21 dgrcl 22393 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
223, 21syl5eqel 2559 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  N  e.  NN0 )
2322nn0red 10853 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  N  e.  RR )
24 nn0re 10804 . . . . . 6  |-  ( M  e.  NN0  ->  M  e.  RR )
25 ltle 9673 . . . . . 6  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <  M  ->  N  <_  M )
)
2623, 24, 25syl2an 477 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  N  <_  M ) )
2726imp 429 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  N  <_  M )
2811, 3dgrub 22394 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0  /\  ( A `
 M )  =/=  0 )  ->  M  <_  N )
29283expia 1198 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  M  <_  N ) )
30 lenlt 9663 . . . . . . . 8  |-  ( ( M  e.  RR  /\  N  e.  RR )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3124, 23, 30syl2anr 478 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( M  <_  N  <->  -.  N  <  M ) )
3229, 31sylibd 214 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  (
( A `  M
)  =/=  0  ->  -.  N  <  M ) )
3332necon4ad 2687 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <  M  ->  ( A `  M )  =  0 ) )
3433imp 429 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( A `  M )  =  0 )
3527, 34jca 532 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <  M )  ->  ( N  <_  M  /\  ( A `  M )  =  0 ) )
3620, 35jaodan 783 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( F  =  0p  \/  N  <  M ) )  ->  ( N  <_  M  /\  ( A `
 M )  =  0 ) )
37 leloe 9671 . . . . . . 7  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M )
) )
3823, 24, 37syl2an 477 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  ->  ( N  <_  M  <->  ( N  <  M  \/  N  =  M ) ) )
3938biimpa 484 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  N  <_  M )  ->  ( N  <  M  \/  N  =  M ) )
4039adantrr 716 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  < 
M  \/  N  =  M ) )
41 fveq2 5866 . . . . . 6  |-  ( N  =  M  ->  ( A `  N )  =  ( A `  M ) )
423, 11dgreq0 22424 . . . . . . . 8  |-  ( F  e.  (Poly `  S
)  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
4342ad2antrr 725 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0p  <->  ( A `  N )  =  0 ) )
44 simprr 756 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( A `  M )  =  0 )
4544eqeq2d 2481 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( ( A `
 N )  =  ( A `  M
)  <->  ( A `  N )  =  0 ) )
4643, 45bitr4d 256 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0p  <->  ( A `  N )  =  ( A `  M ) ) )
4741, 46syl5ibr 221 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( N  =  M  ->  F  = 
0p ) )
4847orim2d 838 . . . 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 388 . 2  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  NN0 )  /\  ( N  <_  M  /\  ( A `  M )  =  0 ) )  ->  ( F  =  0p  \/  N  <  M ) )
5136, 50impbida 830 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {csn 4027   class class class wbr 4447    X. cxp 4997   ` cfv 5588   RRcr 9491   0cc0 9492    < clt 9628    <_ cle 9629   NN0cn0 10795   0pc0p 21839  Polycply 22344  coeffccoe 22346  degcdgr 22347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-inf2 8058  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569  ax-pre-sup 9570  ax-addf 9571
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-of 6524  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-sup 7901  df-oi 7935  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-div 10207  df-nn 10537  df-2 10594  df-3 10595  df-n0 10796  df-z 10865  df-uz 11083  df-rp 11221  df-fz 11673  df-fzo 11793  df-fl 11897  df-seq 12076  df-exp 12135  df-hash 12374  df-cj 12895  df-re 12896  df-im 12897  df-sqrt 13031  df-abs 13032  df-clim 13274  df-rlim 13275  df-sum 13472  df-0p 21840  df-ply 22348  df-coe 22350  df-dgr 22351
This theorem is referenced by:  dgrcolem2  22433  plydivlem4  22454  plydiveu  22456  dgrsub2  30716
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