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Theorem dgradd2 19747
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 19700 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  ( F  o F  +  G
)  e.  (Poly `  CC ) )
213adant3 975 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( F  o F  +  G )  e.  (Poly `  CC )
)
3 dgrcl 19713 . . . . 5  |-  ( ( F  o F  +  G )  e.  (Poly `  CC )  ->  (deg `  ( F  o F  +  G ) )  e.  NN0 )
42, 3syl 15 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  e.  NN0 )
54nn0red 10108 . . 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 19713 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  (deg `  G
)  e.  NN0 )
86, 7syl5eqel 2442 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  N  e.  NN0 )
983ad2ant2 977 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  NN0 )
109nn0red 10108 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  e.  RR )
11 dgradd.1 . . . . . . 7  |-  M  =  (deg `  F )
12 dgrcl 19713 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  (deg `  F
)  e.  NN0 )
1311, 12syl5eqel 2442 . . . . . 6  |-  ( F  e.  (Poly `  S
)  ->  M  e.  NN0 )
14133ad2ant1 976 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  NN0 )
1514nn0red 10108 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  e.  RR )
16 ifcl 3677 . . . 4  |-  ( ( N  e.  RR  /\  M  e.  RR )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1710, 15, 16syl2anc 642 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  e.  RR )
1811, 6dgradd 19746 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
19183adant3 975 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  <_  if ( M  <_  N ,  N ,  M )
)
2010leidd 9426 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  N
)
21 simp3 957 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <  N
)
2215, 10, 21ltled 9054 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  M  <_  N
)
23 breq1 4105 . . . . 5  |-  ( N  =  if ( M  <_  N ,  N ,  M )  ->  ( N  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
24 breq1 4105 . . . . 5  |-  ( M  =  if ( M  <_  N ,  N ,  M )  ->  ( M  <_  N  <->  if ( M  <_  N ,  N ,  M )  <_  N
) )
2523, 24ifboth 3672 . . . 4  |-  ( ( N  <_  N  /\  M  <_  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N )
2620, 22, 25syl2anc 642 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  if ( M  <_  N ,  N ,  M )  <_  N
)
275, 17, 10, 19, 26letrd 9060 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (deg `  ( F  o F  +  G
) )  <_  N
)
28 eqid 2358 . . . . . . . 8  |-  (coeff `  F )  =  (coeff `  F )
29 eqid 2358 . . . . . . . 8  |-  (coeff `  G )  =  (coeff `  G )
3028, 29coeadd 19730 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )
)  ->  (coeff `  ( F  o F  +  G
) )  =  ( (coeff `  F )  o F  +  (coeff `  G ) ) )
31303adant3 975 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  ( F  o F  +  G
) )  =  ( (coeff `  F )  o F  +  (coeff `  G ) ) )
3231fveq1d 5607 . . . . 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 19712 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  (coeff `  F
) : NN0 --> CC )
34333ad2ant1 976 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
) : NN0 --> CC )
35 ffn 5469 . . . . . . . 8  |-  ( (coeff `  F ) : NN0 --> CC 
->  (coeff `  F )  Fn  NN0 )
3634, 35syl 15 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  F
)  Fn  NN0 )
3729coef3 19712 . . . . . . . . 9  |-  ( G  e.  (Poly `  S
)  ->  (coeff `  G
) : NN0 --> CC )
38373ad2ant2 977 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
) : NN0 --> CC )
39 ffn 5469 . . . . . . . 8  |-  ( (coeff `  G ) : NN0 --> CC 
->  (coeff `  G )  Fn  NN0 )
4038, 39syl 15 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  (coeff `  G
)  Fn  NN0 )
41 nn0ex 10060 . . . . . . . 8  |-  NN0  e.  _V
4241a1i 10 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  NN0  e.  _V )
43 inidm 3454 . . . . . . 7  |-  ( NN0 
i^i  NN0 )  =  NN0
4415, 10ltnled 9053 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( M  < 
N  <->  -.  N  <_  M ) )
4521, 44mpbid 201 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  -.  N  <_  M )
46 simp1 955 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  F  e.  (Poly `  S ) )
4728, 11dgrub 19714 . . . . . . . . . . . 12  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0  /\  ( (coeff `  F ) `  N
)  =/=  0 )  ->  N  <_  M
)
48473expia 1153 . . . . . . . . . . 11  |-  ( ( F  e.  (Poly `  S )  /\  N  e.  NN0 )  ->  (
( (coeff `  F
) `  N )  =/=  0  ->  N  <_  M ) )
4946, 9, 48syl2anc 642 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F ) `  N
)  =/=  0  ->  N  <_  M ) )
5049necon1bd 2589 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( -.  N  <_  M  ->  ( (coeff `  F ) `  N
)  =  0 ) )
5145, 50mpd 14 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  F ) `  N
)  =  0 )
5251adantr 451 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  /\  N  e.  NN0 )  ->  ( (coeff `  F ) `  N
)  =  0 )
53 eqidd 2359 . . . . . . 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 6171 . . . . . 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 649 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( ( (coeff `  F )  o F  +  (coeff `  G
) ) `  N
)  =  ( 0  +  ( (coeff `  G ) `  N
) ) )
56 ffvelrn 5743 . . . . . . 7  |-  ( ( (coeff `  G ) : NN0 --> CC  /\  N  e.  NN0 )  ->  (
(coeff `  G ) `  N )  e.  CC )
5738, 9, 56syl2anc 642 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  e.  CC )
5857addid2d 9100 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( 0  +  ( (coeff `  G
) `  N )
)  =  ( (coeff `  G ) `  N
) )
5932, 55, 583eqtrd 2394 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =  ( (coeff `  G ) `  N ) )
60 simp2 956 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  G  e.  (Poly `  S ) )
61 0re 8925 . . . . . . . 8  |-  0  e.  RR
6261a1i 10 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  e.  RR )
6314nn0ge0d 10110 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <_  M
)
6462, 15, 10, 63, 21lelttrd 9061 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  0  <  N
)
6564gt0ne0d 9424 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  =/=  0
)
666, 29dgreq0 19744 . . . . . . 7  |-  ( G  e.  (Poly `  S
)  ->  ( G  =  0 p  <->  ( (coeff `  G ) `  N
)  =  0 ) )
67 fveq2 5605 . . . . . . . 8  |-  ( G  =  0 p  -> 
(deg `  G )  =  (deg `  0 p
) )
68 dgr0 19741 . . . . . . . . 9  |-  (deg ` 
0 p )  =  0
6968eqcomi 2362 . . . . . . . 8  |-  0  =  (deg `  0 p
)
7067, 6, 693eqtr4g 2415 . . . . . . 7  |-  ( G  =  0 p  ->  N  =  0 )
7166, 70syl6bir 220 . . . . . 6  |-  ( G  e.  (Poly `  S
)  ->  ( (
(coeff `  G ) `  N )  =  0  ->  N  =  0 ) )
7271necon3d 2559 . . . . 5  |-  ( G  e.  (Poly `  S
)  ->  ( N  =/=  0  ->  ( (coeff `  G ) `  N
)  =/=  0 ) )
7360, 65, 72sylc 56 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  G ) `  N
)  =/=  0 )
7459, 73eqnetrd 2539 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  ( (coeff `  ( F  o F  +  G ) ) `  N )  =/=  0
)
75 eqid 2358 . . . 4  |-  (coeff `  ( F  o F  +  G ) )  =  (coeff `  ( F  o F  +  G
) )
76 eqid 2358 . . . 4  |-  (deg `  ( F  o F  +  G ) )  =  (deg `  ( F  o F  +  G
) )
7775, 76dgrub 19714 . . 3  |-  ( ( ( F  o F  +  G )  e.  (Poly `  CC )  /\  N  e.  NN0  /\  ( (coeff `  ( F  o F  +  G
) ) `  N
)  =/=  0 )  ->  N  <_  (deg `  ( F  o F  +  G ) ) )
782, 9, 74, 77syl3anc 1182 . 2  |-  ( ( F  e.  (Poly `  S )  /\  G  e.  (Poly `  S )  /\  M  <  N )  ->  N  <_  (deg `  ( F  o F  +  G ) ) )
795, 10letri3d 9048 . 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 ) ) ) ) )
8027, 78, 79mpbir2and 888 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 358    /\ w3a 934    = wceq 1642    e. wcel 1710    =/= wne 2521   _Vcvv 2864   ifcif 3641   class class class wbr 4102    Fn wfn 5329   -->wf 5330   ` cfv 5334  (class class class)co 5942    o Fcof 6160   CCcc 8822   RRcr 8823   0cc0 8824    + caddc 8827    < clt 8954    <_ cle 8955   NN0cn0 10054   0 pc0p 19122  Polycply 19664  coeffccoe 19666  degcdgr 19667
This theorem is referenced by:  dgrcolem2  19753  plyremlem  19782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-inf2 7429  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901  ax-pre-sup 8902  ax-addf 8903
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-int 3942  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-se 4432  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-isom 5343  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-of 6162  df-1st 6206  df-2nd 6207  df-riota 6388  df-recs 6472  df-rdg 6507  df-1o 6563  df-oadd 6567  df-er 6744  df-map 6859  df-pm 6860  df-en 6949  df-dom 6950  df-sdom 6951  df-fin 6952  df-sup 7281  df-oi 7312  df-card 7659  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-div 9511  df-nn 9834  df-2 9891  df-3 9892  df-n0 10055  df-z 10114  df-uz 10320  df-rp 10444  df-fz 10872  df-fzo 10960  df-fl 11014  df-seq 11136  df-exp 11195  df-hash 11428  df-cj 11674  df-re 11675  df-im 11676  df-sqr 11810  df-abs 11811  df-clim 12052  df-rlim 12053  df-sum 12250  df-0p 19123  df-ply 19668  df-coe 19670  df-dgr 19671
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