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Theorem dgradd2 20191
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  o F  +  G
) )  =  N )

Proof of Theorem dgradd2
StepHypRef Expression
1 plyaddcl 20144 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  G
)  e.  (Poly `  CC ) )
213adant3 978 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( F  o F  +  G )  e.  (Poly `  CC )
)
3 dgrcl 20157 . . . . 5  |-  ( ( F  o F  +  G )  e.  (Poly `  CC )  ->  (deg `  ( F  o F  +  G ) )  e.  NN0 )
42, 3syl 16 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  e.  NN0 )
54nn0red 10280 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  e.  RR )
6 dgradd.2 . . . . . . 7  |-  N  =  (deg `  G )
7 dgrcl 20157 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
86, 7syl5eqel 2522 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
983ad2ant2 980 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  NN0 )
109nn0red 10280 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  RR )
11 dgradd.1 . . . . . . 7  |-  M  =  (deg `  F )
12 dgrcl 20157 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
1311, 12syl5eqel 2522 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
14133ad2ant1 979 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  NN0 )
1514nn0red 10280 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  RR )
16 ifcl 3777 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1710, 15, 16syl2anc 644 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1811, 6dgradd 20190 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
19183adant3 978 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
2010leidd 9598 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  N
)
21 simp3 960 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <  N
)
2215, 10, 21ltled 9226 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <_  N
)
23 breq1 4218 . . . . 5  |-  ( N  =  if ( M  <_  N ,  N ,  M )  ->  ( N  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
24 breq1 4218 . . . . 5  |-  ( M  =  if ( M  <_  N ,  N ,  M )  ->  ( M  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
2523, 24ifboth 3772 . . . 4  |-  ( ( N  <_  N  /\  M  <_  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N )
2620, 22, 25syl2anc 644 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N
)
275, 17, 10, 19, 26letrd 9232 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  <_  N
)
28 eqid 2438 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
29 eqid 2438 . . . . . . . 8  |-  (coeff `  G )  =  (coeff `  G )
3028, 29coeadd 20174 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  G
) )  =  ( (coeff `  F )  o F  +  (coeff `  G ) ) )
31303adant3 978 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  ( F  o F  +  G
) )  =  ( (coeff `  F )  o F  +  (coeff `  G ) ) )
3231fveq1d 5733 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =  ( ( (coeff `  F
)  o F  +  (coeff `  G ) ) `
 N ) )
3328coef3 20156 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
34333ad2ant1 979 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
) : NN0 --> CC )
35 ffn 5594 . . . . . . . 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 20156 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
38373ad2ant2 980 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
) : NN0 --> CC )
39 ffn 5594 . . . . . . . 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 10232 . . . . . . . 8  |-  NN0  e.  _V
4241a1i 11 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  NN0  e.  _V )
43 inidm 3552 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
4415, 10ltnled 9225 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( M  < 
N  <->  -.  N  <_  M ) )
4521, 44mpbid 203 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  -.  N  <_  M )
46 simp1 958 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  F  e.  (Poly `  S ) )
4728, 11dgrub 20158 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  M
)
48473expia 1156 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  (
( (coeff `  F
) `  N )  =/=  0  ->  N  <_  M ) )
4946, 9, 48syl2anc 644 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F ) `  N
)  =/=  0  ->  N  <_  M ) )
5049necon1bd 2674 . . . . . . . . 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 453 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  F ) `  N
)  =  0 )
53 eqidd 2439 . . . . . . 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 6317 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( ( (coeff `  F )  o F  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
559, 54mpdan 651 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F )  o F  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
5638, 9ffvelrnd 5874 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  e.  CC )
5756addid2d 9272 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( 0  +  ( (coeff `  G
) `  N )
)  =  ( (coeff `  G ) `  N
) )
5832, 55, 573eqtrd 2474 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =  ( (coeff `  G ) `  N ) )
59 simp2 959 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  G  e.  (Poly `  S ) )
60 0re 9096 . . . . . . . 8  |-  0  e.  RR
6160a1i 11 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  e.  RR )
6214nn0ge0d 10282 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <_  M
)
6361, 15, 10, 62, 21lelttrd 9233 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <  N
)
6463gt0ne0d 9596 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  =/=  0
)
656, 29dgreq0 20188 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
66 fveq2 5731 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
67 dgr0 20185 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
6867eqcomi 2442 . . . . . . . 8  |-  0  =  (deg `  0 p
)
6966, 6, 683eqtr4g 2495 . . . . . . 7  |-  ( G  =  0 p  ->  N  =  0 )
7065, 69syl6bir 222 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  ( (
(coeff `  G ) `  N )  =  0  ->  N  =  0 ) )
7170necon3d 2641 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( N  =/=  0  ->  ( (coeff `  G ) `  N
)  =/=  0 ) )
7259, 64, 71sylc 59 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
7358, 72eqnetrd 2621 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =/=  0
)
74 eqid 2438 . . . 4  |-  (coeff `  ( F  o F  +  G ) )  =  (coeff `  ( F  o F  +  G
) )
75 eqid 2438 . . . 4  |-  (deg `  ( F  o F  +  G ) )  =  (deg `  ( F  o F  +  G
) )
7674, 75dgrub 20158 . . 3  |-  ( ( ( F  o F  +  G )  e.  (Poly `  CC )  /\  N  e.  NN0  /\  ( (coeff `  ( F  o F  +  G
) ) `  N
)  =/=  0 )  ->  N  <_  (deg `  ( F  o F  +  G ) ) )
772, 9, 73, 76syl3anc 1185 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  (deg `  ( F  o F  +  G ) ) )
785, 10letri3d 9220 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (deg `  ( F  o F  +  G ) )  =  N  <->  ( (deg `  ( F  o F  +  G ) )  <_  N  /\  N  <_  (deg `  ( F  o F  +  G ) ) ) ) )
7927, 77, 78mpbir2and 890 1  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  =  N )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2601   _Vcvv 2958   ifcif 3741   class class class wbr 4215    Fn wfn 5452   -->wf 5453   ` cfv 5457  (class class class)co 6084    o Fcof 6306   CCcc 8993   RRcr 8994   0cc0 8995    + caddc 8998    < clt 9125    <_ cle 9126   NN0cn0 10226   0 pc0p 19564  Polycply 20108  coeffccoe 20110  degcdgr 20111
This theorem is referenced by:  dgrcolem2  20197  plyremlem  20226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073  ax-addf 9074
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-of 6308  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-fz 11049  df-fzo 11141  df-fl 11207  df-seq 11329  df-exp 11388  df-hash 11624  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-rlim 12288  df-sum 12485  df-0p 19565  df-ply 20112  df-coe 20114  df-dgr 20115
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