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Theorem dgradd2 22532
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 22485 . . . . . 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 22498 . . . . 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 10865 . . 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 22498 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
86, 7syl5eqel 2559 . . . . . 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 10865 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  RR )
11 dgradd.1 . . . . . . 7  |-  M  =  (deg `  F )
12 dgrcl 22498 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
1311, 12syl5eqel 2559 . . . . . 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 10865 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  RR )
16 ifcl 3987 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1710, 15, 16syl2anc 661 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1811, 6dgradd 22531 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
19183adant3 1016 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
2010leidd 10131 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  N
)
21 simp3 998 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <  N
)
2215, 10, 21ltled 9744 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <_  N
)
23 breq1 4456 . . . . 5  |-  ( N  =  if ( M  <_  N ,  N ,  M )  ->  ( N  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
24 breq1 4456 . . . . 5  |-  ( M  =  if ( M  <_  N ,  N ,  M )  ->  ( M  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
2523, 24ifboth 3981 . . . 4  |-  ( ( N  <_  N  /\  M  <_  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N )
2620, 22, 25syl2anc 661 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N
)
275, 17, 10, 19, 26letrd 9750 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  oF  +  G
) )  <_  N
)
28 eqid 2467 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
29 eqid 2467 . . . . . . . 8  |-  (coeff `  G )  =  (coeff `  G )
3028, 29coeadd 22515 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  oF  +  G
) )  =  ( (coeff `  F )  oF  +  (coeff `  G ) ) )
31303adant3 1016 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  ( F  oF  +  G
) )  =  ( (coeff `  F )  oF  +  (coeff `  G ) ) )
3231fveq1d 5874 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  oF  +  G ) ) `  N )  =  ( ( (coeff `  F
)  oF  +  (coeff `  G ) ) `
 N ) )
3328coef3 22497 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
34333ad2ant1 1017 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
) : NN0 --> CC )
35 ffn 5737 . . . . . . . 8  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
3634, 35syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
)  Fn  NN0 )
3729coef3 22497 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
38373ad2ant2 1018 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
) : NN0 --> CC )
39 ffn 5737 . . . . . . . 8  |-  ( (coeff `  G ) : NN0 --> CC 
->  (coeff `  G )  Fn  NN0 )
4038, 39syl 16 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
)  Fn  NN0 )
41 nn0ex 10813 . . . . . . . 8  |-  NN0  e.  _V
4241a1i 11 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  NN0  e.  _V )
43 inidm 3712 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
4415, 10ltnled 9743 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( M  < 
N  <->  -.  N  <_  M ) )
4521, 44mpbid 210 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  -.  N  <_  M )
46 simp1 996 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  F  e.  (Poly `  S ) )
4728, 11dgrub 22499 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  M
)
48473expia 1198 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  (
( (coeff `  F
) `  N )  =/=  0  ->  N  <_  M ) )
4946, 9, 48syl2anc 661 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F ) `  N
)  =/=  0  ->  N  <_  M ) )
5049necon1bd 2685 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( -.  N  <_  M  ->  ( (coeff `  F ) `  N
)  =  0 ) )
5145, 50mpd 15 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  F ) `  N
)  =  0 )
5251adantr 465 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  F ) `  N
)  =  0 )
53 eqidd 2468 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  G ) `  N
)  =  ( (coeff `  G ) `  N
) )
5436, 40, 42, 42, 43, 52, 53ofval 6544 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( ( (coeff `  F )  oF  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
559, 54mpdan 668 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F )  oF  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
5638, 9ffvelrnd 6033 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  e.  CC )
5756addid2d 9792 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( 0  +  ( (coeff `  G
) `  N )
)  =  ( (coeff `  G ) `  N
) )
5832, 55, 573eqtrd 2512 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  oF  +  G ) ) `  N )  =  ( (coeff `  G ) `  N ) )
59 simp2 997 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  G  e.  (Poly `  S ) )
60 0red 9609 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  e.  RR )
6114nn0ge0d 10867 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <_  M
)
6260, 15, 10, 61, 21lelttrd 9751 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <  N
)
6362gt0ne0d 10129 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  =/=  0
)
646, 29dgreq0 22529 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
65 fveq2 5872 . . . . . . . 8  |-  ( G  =  0p  -> 
(deg `  G )  =  (deg `  0p
) )
66 dgr0 22526 . . . . . . . . 9  |-  (deg ` 
0p )  =  0
6766eqcomi 2480 . . . . . . . 8  |-  0  =  (deg `  0p
)
6865, 6, 673eqtr4g 2533 . . . . . . 7  |-  ( G  =  0p  ->  N  =  0 )
6964, 68syl6bir 229 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  ( (
(coeff `  G ) `  N )  =  0  ->  N  =  0 ) )
7069necon3d 2691 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( N  =/=  0  ->  ( (coeff `  G ) `  N
)  =/=  0 ) )
7159, 63, 70sylc 60 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
7258, 71eqnetrd 2760 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  oF  +  G ) ) `  N )  =/=  0
)
73 eqid 2467 . . . 4  |-  (coeff `  ( F  oF  +  G ) )  =  (coeff `  ( F  oF  +  G
) )
74 eqid 2467 . . . 4  |-  (deg `  ( F  oF  +  G ) )  =  (deg `  ( F  oF  +  G
) )
7573, 74dgrub 22499 . . 3  |-  ( ( ( F  oF  +  G )  e.  (Poly `  CC )  /\  N  e.  NN0  /\  ( (coeff `  ( F  oF  +  G
) ) `  N
)  =/=  0 )  ->  N  <_  (deg `  ( F  oF  +  G ) ) )
762, 9, 72, 75syl3anc 1228 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  (deg `  ( F  oF  +  G ) ) )
775, 10letri3d 9738 . 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 ) ) ) ) )
7827, 76, 77mpbir2and 920 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 1379    e. wcel 1767    =/= wne 2662   _Vcvv 3118   ifcif 3945   class class class wbr 4453    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6295    oFcof 6533   CCcc 9502   RRcr 9503   0cc0 9504    + caddc 9507    < clt 9640    <_ cle 9641   NN0cn0 10807   0pc0p 21944  Polycply 22449  coeffccoe 22451  degcdgr 22452
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-0p 21945  df-ply 22453  df-coe 22455  df-dgr 22456
This theorem is referenced by:  dgrcolem2  22538  plyremlem  22567
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