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Theorem dgradd2 22791
Description: The degree of a sum of polynomials of unequal degrees is the degree of the larger polynomial. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
dgradd.1  |-  M  =  (deg `  F )
dgradd.2  |-  N  =  (deg `  G )
Assertion
Ref Expression
dgradd2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  =  N )

Proof of Theorem dgradd2
StepHypRef Expression
1 plyaddcl 22743 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  oF  +  G
)  e.  (Poly `  CC ) )
213adant3 1016 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( F  oF  +  G )  e.  (Poly `  CC )
)
3 dgrcl 22756 . . . . 5  |-  ( ( F  oF  +  G )  e.  (Poly `  CC )  ->  (deg `  ( F  oF  +  G ) )  e.  NN0 )
42, 3syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  e.  NN0 )
54nn0red 10874 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  e.  RR )
6 dgradd.2 . . . . . . 7  |-  N  =  (deg `  G )
7 dgrcl 22756 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
86, 7syl5eqel 2549 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
983ad2ant2 1018 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  NN0 )
109nn0red 10874 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  RR )
11 dgradd.1 . . . . . . 7  |-  M  =  (deg `  F )
12 dgrcl 22756 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
1311, 12syl5eqel 2549 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
14133ad2ant1 1017 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  NN0 )
1514nn0red 10874 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  RR )
1610, 15ifcld 3987 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1711, 6dgradd 22790 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
18173adant3 1016 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
1910leidd 10140 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  N
)
20 simp3 998 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <  N
)
2115, 10, 20ltled 9750 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <_  N
)
22 breq1 4459 . . . . 5  |-  ( N  =  if ( M  <_  N ,  N ,  M )  ->  ( N  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
23 breq1 4459 . . . . 5  |-  ( M  =  if ( M  <_  N ,  N ,  M )  ->  ( M  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
2422, 23ifboth 3980 . . . 4  |-  ( ( N  <_  N  /\  M  <_  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N )
2519, 21, 24syl2anc 661 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N
)
265, 16, 10, 18, 25letrd 9756 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  <_  N
)
27 eqid 2457 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
28 eqid 2457 . . . . . . . 8  |-  (coeff `  G )  =  (coeff `  G )
2927, 28coeadd 22774 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  G
) )  =  ( (coeff `  F )  oF  +  (coeff `  G ) ) )
30293adant3 1016 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  ( F  oF  +  G
) )  =  ( (coeff `  F )  oF  +  (coeff `  G ) ) )
3130fveq1d 5874 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  oF  +  G ) ) `  N )  =  ( ( (coeff `  F
)  oF  +  (coeff `  G ) ) `
 N ) )
3227coef3 22755 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
33323ad2ant1 1017 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
) : NN0 --> CC )
34 ffn 5737 . . . . . . . 8  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
3533, 34syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
)  Fn  NN0 )
3628coef3 22755 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
37363ad2ant2 1018 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
) : NN0 --> CC )
38 ffn 5737 . . . . . . . 8  |-  ( (coeff `  G ) : NN0 --> CC 
->  (coeff `  G )  Fn  NN0 )
3937, 38syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
)  Fn  NN0 )
40 nn0ex 10822 . . . . . . . 8  |-  NN0  e.  _V
4140a1i 11 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  NN0  e.  _V )
42 inidm 3703 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
4315, 10ltnled 9749 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( M  < 
N  <->  -.  N  <_  M ) )
4420, 43mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  -.  N  <_  M )
45 simp1 996 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  F  e.  (Poly `  S ) )
4627, 11dgrub 22757 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  M
)
47463expia 1198 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  (
( (coeff `  F
) `  N )  =/=  0  ->  N  <_  M ) )
4845, 9, 47syl2anc 661 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F ) `  N
)  =/=  0  ->  N  <_  M ) )
4948necon1bd 2675 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( -.  N  <_  M  ->  ( (coeff `  F ) `  N
)  =  0 ) )
5044, 49mpd 15 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  F ) `  N
)  =  0 )
5150adantr 465 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  F ) `  N
)  =  0 )
52 eqidd 2458 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  G ) `  N
)  =  ( (coeff `  G ) `  N
) )
5335, 39, 41, 41, 42, 51, 52ofval 6548 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( ( (coeff `  F )  oF  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
549, 53mpdan 668 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F )  oF  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
5537, 9ffvelrnd 6033 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  e.  CC )
5655addid2d 9798 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( 0  +  ( (coeff `  G
) `  N )
)  =  ( (coeff `  G ) `  N
) )
5731, 54, 563eqtrd 2502 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  oF  +  G ) ) `  N )  =  ( (coeff `  G ) `  N ) )
58 simp2 997 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  G  e.  (Poly `  S ) )
59 0red 9614 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  e.  RR )
6014nn0ge0d 10876 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <_  M
)
6159, 15, 10, 60, 20lelttrd 9757 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <  N
)
6261gt0ne0d 10138 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  =/=  0
)
636, 28dgreq0 22788 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
64 fveq2 5872 . . . . . . . 8  |-  ( G  =  0p  -> 
(deg `  G )  =  (deg `  0p
) )
65 dgr0 22785 . . . . . . . . 9  |-  (deg ` 
0p )  =  0
6665eqcomi 2470 . . . . . . . 8  |-  0  =  (deg `  0p
)
6764, 6, 663eqtr4g 2523 . . . . . . 7  |-  ( G  =  0p  ->  N  =  0 )
6863, 67syl6bir 229 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  ( (
(coeff `  G ) `  N )  =  0  ->  N  =  0 ) )
6968necon3d 2681 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( N  =/=  0  ->  ( (coeff `  G ) `  N
)  =/=  0 ) )
7058, 62, 69sylc 60 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
7157, 70eqnetrd 2750 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  oF  +  G ) ) `  N )  =/=  0
)
72 eqid 2457 . . . 4  |-  (coeff `  ( F  oF  +  G ) )  =  (coeff `  ( F  oF  +  G
) )
73 eqid 2457 . . . 4  |-  (deg `  ( F  oF  +  G ) )  =  (deg `  ( F  oF  +  G
) )
7472, 73dgrub 22757 . . 3  |-  ( ( ( F  oF  +  G )  e.  (Poly `  CC )  /\  N  e.  NN0  /\  ( (coeff `  ( F  oF  +  G
) ) `  N
)  =/=  0 )  ->  N  <_  (deg `  ( F  oF  +  G ) ) )
752, 9, 71, 74syl3anc 1228 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  (deg `  ( F  oF  +  G ) ) )
765, 10letri3d 9744 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (deg `  ( F  oF  +  G ) )  =  N  <->  ( (deg `  ( F  oF  +  G ) )  <_  N  /\  N  <_  (deg `  ( F  oF  +  G ) ) ) ) )
7726, 75, 76mpbir2and 922 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  =  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   _Vcvv 3109   ifcif 3944   class class class wbr 4456    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296    oFcof 6537   CCcc 9507   RRcr 9508   0cc0 9509    + caddc 9512    < clt 9645    <_ cle 9646   NN0cn0 10816   0pc0p 22202  Polycply 22707  coeffccoe 22709  degcdgr 22710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586  ax-pre-sup 9587  ax-addf 9588
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-of 6539  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-map 7440  df-pm 7441  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-sup 7919  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-div 10228  df-nn 10557  df-2 10615  df-3 10616  df-n0 10817  df-z 10886  df-uz 11107  df-rp 11246  df-fz 11698  df-fzo 11822  df-fl 11932  df-seq 12111  df-exp 12170  df-hash 12409  df-cj 12944  df-re 12945  df-im 12946  df-sqrt 13080  df-abs 13081  df-clim 13323  df-rlim 13324  df-sum 13521  df-0p 22203  df-ply 22711  df-coe 22713  df-dgr 22714
This theorem is referenced by:  dgrcolem2  22797  plyremlem  22826
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