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Theorem dgradd2 20143
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 20096 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  G
)  e.  (Poly `  CC ) )
213adant3 977 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( F  o F  +  G )  e.  (Poly `  CC )
)
3 dgrcl 20109 . . . . 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 10235 . . 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 20109 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
86, 7syl5eqel 2492 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
983ad2ant2 979 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  NN0 )
109nn0red 10235 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  RR )
11 dgradd.1 . . . . . . 7  |-  M  =  (deg `  F )
12 dgrcl 20109 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
1311, 12syl5eqel 2492 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
14133ad2ant1 978 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  NN0 )
1514nn0red 10235 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  RR )
16 ifcl 3739 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1710, 15, 16syl2anc 643 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1811, 6dgradd 20142 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
19183adant3 977 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
2010leidd 9553 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  N
)
21 simp3 959 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <  N
)
2215, 10, 21ltled 9181 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <_  N
)
23 breq1 4179 . . . . 5  |-  ( N  =  if ( M  <_  N ,  N ,  M )  ->  ( N  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
24 breq1 4179 . . . . 5  |-  ( M  =  if ( M  <_  N ,  N ,  M )  ->  ( M  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
2523, 24ifboth 3734 . . . 4  |-  ( ( N  <_  N  /\  M  <_  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N )
2620, 22, 25syl2anc 643 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N
)
275, 17, 10, 19, 26letrd 9187 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  <_  N
)
28 eqid 2408 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
29 eqid 2408 . . . . . . . 8  |-  (coeff `  G )  =  (coeff `  G )
3028, 29coeadd 20126 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  G
) )  =  ( (coeff `  F )  o F  +  (coeff `  G ) ) )
31303adant3 977 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  ( F  o F  +  G
) )  =  ( (coeff `  F )  o F  +  (coeff `  G ) ) )
3231fveq1d 5693 . . . . 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 20108 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
34333ad2ant1 978 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
) : NN0 --> CC )
35 ffn 5554 . . . . . . . 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 20108 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
38373ad2ant2 979 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
) : NN0 --> CC )
39 ffn 5554 . . . . . . . 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 10187 . . . . . . . 8  |-  NN0  e.  _V
4241a1i 11 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  NN0  e.  _V )
43 inidm 3514 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
4415, 10ltnled 9180 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( M  < 
N  <->  -.  N  <_  M ) )
4521, 44mpbid 202 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  -.  N  <_  M )
46 simp1 957 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  F  e.  (Poly `  S ) )
4728, 11dgrub 20110 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  M
)
48473expia 1155 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  (
( (coeff `  F
) `  N )  =/=  0  ->  N  <_  M ) )
4946, 9, 48syl2anc 643 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F ) `  N
)  =/=  0  ->  N  <_  M ) )
5049necon1bd 2639 . . . . . . . . 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 452 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  F ) `  N
)  =  0 )
53 eqidd 2409 . . . . . . 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 6277 . . . . . 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 650 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F )  o F  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
5638, 9ffvelrnd 5834 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  e.  CC )
5756addid2d 9227 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( 0  +  ( (coeff `  G
) `  N )
)  =  ( (coeff `  G ) `  N
) )
5832, 55, 573eqtrd 2444 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =  ( (coeff `  G ) `  N ) )
59 simp2 958 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  G  e.  (Poly `  S ) )
60 0re 9051 . . . . . . . 8  |-  0  e.  RR
6160a1i 11 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  e.  RR )
6214nn0ge0d 10237 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <_  M
)
6361, 15, 10, 62, 21lelttrd 9188 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <  N
)
6463gt0ne0d 9551 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  =/=  0
)
656, 29dgreq0 20140 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
66 fveq2 5691 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
67 dgr0 20137 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
6867eqcomi 2412 . . . . . . . 8  |-  0  =  (deg `  0 p
)
6966, 6, 683eqtr4g 2465 . . . . . . 7  |-  ( G  =  0 p  ->  N  =  0 )
7065, 69syl6bir 221 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  ( (
(coeff `  G ) `  N )  =  0  ->  N  =  0 ) )
7170necon3d 2609 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( N  =/=  0  ->  ( (coeff `  G ) `  N
)  =/=  0 ) )
7259, 64, 71sylc 58 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
7358, 72eqnetrd 2589 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =/=  0
)
74 eqid 2408 . . . 4  |-  (coeff `  ( F  o F  +  G ) )  =  (coeff `  ( F  o F  +  G
) )
75 eqid 2408 . . . 4  |-  (deg `  ( F  o F  +  G ) )  =  (deg `  ( F  o F  +  G
) )
7674, 75dgrub 20110 . . 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 1184 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  (deg `  ( F  o F  +  G ) ) )
785, 10letri3d 9175 . 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 889 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920   ifcif 3703   class class class wbr 4176    Fn wfn 5412   -->wf 5413   ` cfv 5417  (class class class)co 6044    o Fcof 6266   CCcc 8948   RRcr 8949   0cc0 8950    + caddc 8953    < clt 9080    <_ cle 9081   NN0cn0 10181   0 pc0p 19518  Polycply 20060  coeffccoe 20062  degcdgr 20063
This theorem is referenced by:  dgrcolem2  20149  plyremlem  20178
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-inf2 7556  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027  ax-pre-sup 9028  ax-addf 9029
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-se 4506  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-isom 5426  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-of 6268  df-1st 6312  df-2nd 6313  df-riota 6512  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-map 6983  df-pm 6984  df-en 7073  df-dom 7074  df-sdom 7075  df-fin 7076  df-sup 7408  df-oi 7439  df-card 7786  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-div 9638  df-nn 9961  df-2 10018  df-3 10019  df-n0 10182  df-z 10243  df-uz 10449  df-rp 10573  df-fz 11004  df-fzo 11095  df-fl 11161  df-seq 11283  df-exp 11342  df-hash 11578  df-cj 11863  df-re 11864  df-im 11865  df-sqr 11999  df-abs 12000  df-clim 12241  df-rlim 12242  df-sum 12439  df-0p 19519  df-ply 20064  df-coe 20066  df-dgr 20067
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