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

Step	Hyp	Ref	Expression
1		plyaddcl 20096	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F o F + G ) e. (Poly ` CC ) )
2	1	3adant3 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 )
4	2, 3	syl 16	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F o F + G ) ) e. NN0 )
5	4	nn0red 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 )
8	6, 7	syl5eqel 2492	. . . . . 6 |- ( G e. (Poly ` S ) -> N e. NN0 )
9	8	3ad2ant2 979	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N e. NN0 )
10	9	nn0red 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 )
13	11, 12	syl5eqel 2492	. . . . . 6 |- ( F e. (Poly ` S ) -> M e. NN0 )
14	13	3ad2ant1 978	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> M e. NN0 )
15	14	nn0red 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 )
17	10, 15, 16	syl2anc 643	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> if ( M <_ N , N , M ) e. RR )
18	11, 6	dgradd 20142	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` ( F o F + G ) ) <_ if ( M <_ N , N , M ) )
19	18	3adant3 977	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F o F + G ) ) <_ if ( M <_ N , N , M ) )
20	10	leidd 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 )
22	15, 10, 21	ltled 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 ) )
25	23, 24	ifboth 3734	. . . 4 |- ( ( N <_ N /\ M <_ N ) -> if ( M <_ N , N , M ) <_ N )
26	20, 22, 25	syl2anc 643	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> if ( M <_ N , N , M ) <_ N )
27	5, 17, 10, 19, 26	letrd 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 )
30	28, 29	coeadd 20126	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` ( F o F + G ) ) = ( (coeff ` F ) o F + (coeff ` G ) ) )
31	30	3adant3 977	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` ( F o F + G ) ) = ( (coeff ` F ) o F + (coeff ` G ) ) )
32	31	fveq1d 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 ) )
33	28	coef3 20108	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
34	33	3ad2ant1 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 )
36	34, 35	syl 16	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` F ) Fn NN0 )
37	29	coef3 20108	. . . . . . . . 9 |- ( G e. (Poly ` S ) -> (coeff ` G ) : NN0 --> CC )
38	37	3ad2ant2 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 )
40	38, 39	syl 16	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` G ) Fn NN0 )
41		nn0ex 10187	. . . . . . . 8 |- NN0 e. _V
42	41	a1i 11	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> NN0 e. _V )
43		inidm 3514	. . . . . . 7 |- ( NN0 i^i NN0 ) = NN0
44	15, 10	ltnled 9180	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( M < N <-> -. N <_ M ) )
45	21, 44	mpbid 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 ) )
47	28, 11	dgrub 20110	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ N e. NN0 /\ ( (coeff ` F ) ` N ) =/= 0 ) -> N <_ M )
48	47	3expia 1155	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ N e. NN0 ) -> ( ( (coeff ` F ) ` N ) =/= 0 -> N <_ M ) )
49	46, 9, 48	syl2anc 643	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( ( (coeff ` F ) ` N ) =/= 0 -> N <_ M ) )
50	49	necon1bd 2639	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( -. N <_ M -> ( (coeff ` F ) ` N ) = 0 ) )
51	45, 50	mpd 15	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` F ) ` N ) = 0 )
52	51	adantr 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 ) )
54	36, 40, 42, 42, 43, 52, 53	ofval 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 ) ) )
55	9, 54	mpdan 650	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( ( (coeff ` F ) o F + (coeff ` G ) ) ` N ) = ( 0 + ( (coeff ` G ) ` N ) ) )
56	38, 9	ffvelrnd 5834	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` G ) ` N ) e. CC )
57	56	addid2d 9227	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( 0 + ( (coeff ` G ) ` N ) ) = ( (coeff ` G ) ` N ) )
58	32, 55, 57	3eqtrd 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
61	60	a1i 11	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> 0 e. RR )
62	14	nn0ge0d 10237	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> 0 <_ M )
63	61, 15, 10, 62, 21	lelttrd 9188	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> 0 < N )
64	63	gt0ne0d 9551	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N =/= 0 )
65	6, 29	dgreq0 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
68	67	eqcomi 2412	. . . . . . . 8 |- 0 = (deg ` 0 p )
69	66, 6, 68	3eqtr4g 2465	. . . . . . 7 |- ( G = 0 p -> N = 0 )
70	65, 69	syl6bir 221	. . . . . 6 |- ( G e. (Poly ` S ) -> ( ( (coeff ` G ) ` N ) = 0 -> N = 0 ) )
71	70	necon3d 2609	. . . . 5 |- ( G e. (Poly ` S ) -> ( N =/= 0 -> ( (coeff ` G ) ` N ) =/= 0 ) )
72	59, 64, 71	sylc 58	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` G ) ` N ) =/= 0 )
73	58, 72	eqnetrd 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 ) )
76	74, 75	dgrub 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 ) ) )
77	2, 9, 73, 76	syl3anc 1184	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N <_ (deg ` ( F o F + G ) ) )
78	5, 10	letri3d 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 ) ) ) ) )
79	27, 77, 78	mpbir2and 889	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F o F + G ) ) = N )

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