Theorem dgradd2 24024

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

Step	Hyp	Ref	Expression
1		plyaddcl 23976	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF + G ) e. (Poly ` CC ) )
2	1	3adant3 1081	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( F oF + G ) e. (Poly ` CC ) )
3		dgrcl 23989	. . . . 5 |- ( ( F oF + G ) e. (Poly ` CC ) -> (deg ` ( F oF + G ) ) e. NN0 )
4	2, 3	syl 17	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F oF + G ) ) e. NN0 )
5	4	nn0red 11352	. . 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 23989	. . . . . . 7 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
8	6, 7	syl5eqel 2705	. . . . . 6 |- ( G e. (Poly ` S ) -> N e. NN0 )
9	8	3ad2ant2 1083	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N e. NN0 )
10	9	nn0red 11352	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N e. RR )
11		dgradd.1	. . . . . . 7 |- M = (deg ` F )
12		dgrcl 23989	. . . . . . 7 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
13	11, 12	syl5eqel 2705	. . . . . 6 |- ( F e. (Poly ` S ) -> M e. NN0 )
14	13	3ad2ant1 1082	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> M e. NN0 )
15	14	nn0red 11352	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> M e. RR )
16	10, 15	ifcld 4131	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> if ( M <_ N , N , M ) e. RR )
17	11, 6	dgradd 24023	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) )
18	17	3adant3 1081	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) )
19	10	leidd 10594	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N <_ N )
20		simp3 1063	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> M < N )
21	15, 10, 20	ltled 10185	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> M <_ N )
22		breq1 4656	. . . . 5 |- ( N = if ( M <_ N , N , M ) -> ( N <_ N <-> if ( M <_ N , N , M ) <_ N ) )
23		breq1 4656	. . . . 5 |- ( M = if ( M <_ N , N , M ) -> ( M <_ N <-> if ( M <_ N , N , M ) <_ N ) )
24	22, 23	ifboth 4124	. . . 4 |- ( ( N <_ N /\ M <_ N ) -> if ( M <_ N , N , M ) <_ N )
25	19, 21, 24	syl2anc 693	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> if ( M <_ N , N , M ) <_ N )
26	5, 16, 10, 18, 25	letrd 10194	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F oF + G ) ) <_ N )
27		eqid 2622	. . . . . . . 8 |- (coeff ` F ) = (coeff ` F )
28		eqid 2622	. . . . . . . 8 |- (coeff ` G ) = (coeff ` G )
29	27, 28	coeadd 24007	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` ( F oF + G ) ) = ( (coeff ` F ) oF + (coeff ` G ) ) )
30	29	3adant3 1081	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` ( F oF + G ) ) = ( (coeff ` F ) oF + (coeff ` G ) ) )
31	30	fveq1d 6193	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` ( F oF + G ) ) ` N ) = ( ( (coeff ` F ) oF + (coeff ` G ) ) ` N ) )
32	27	coef3 23988	. . . . . . . . 9 |- ( F e. (Poly ` S ) -> (coeff ` F ) : NN0 --> CC )
33	32	3ad2ant1 1082	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` F ) : NN0 --> CC )
34		ffn 6045	. . . . . . . 8 |- ( (coeff ` F ) : NN0 --> CC -> (coeff ` F ) Fn NN0 )
35	33, 34	syl 17	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` F ) Fn NN0 )
36	28	coef3 23988	. . . . . . . . 9 |- ( G e. (Poly ` S ) -> (coeff ` G ) : NN0 --> CC )
37	36	3ad2ant2 1083	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` G ) : NN0 --> CC )
38		ffn 6045	. . . . . . . 8 |- ( (coeff ` G ) : NN0 --> CC -> (coeff ` G ) Fn NN0 )
39	37, 38	syl 17	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (coeff ` G ) Fn NN0 )
40		nn0ex 11298	. . . . . . . 8 |- NN0 e. _V
41	40	a1i 11	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> NN0 e. _V )
42		inidm 3822	. . . . . . 7 |- ( NN0 i^i NN0 ) = NN0
43	15, 10	ltnled 10184	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( M < N <-> -. N <_ M ) )
44	20, 43	mpbid 222	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> -. N <_ M )
45		simp1 1061	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> F e. (Poly ` S ) )
46	27, 11	dgrub 23990	. . . . . . . . . . . 12 |- ( ( F e. (Poly ` S ) /\ N e. NN0 /\ ( (coeff ` F ) ` N ) =/= 0 ) -> N <_ M )
47	46	3expia 1267	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ N e. NN0 ) -> ( ( (coeff ` F ) ` N ) =/= 0 -> N <_ M ) )
48	45, 9, 47	syl2anc 693	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( ( (coeff ` F ) ` N ) =/= 0 -> N <_ M ) )
49	48	necon1bd 2812	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( -. N <_ M -> ( (coeff ` F ) ` N ) = 0 ) )
50	44, 49	mpd 15	. . . . . . . 8 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` F ) ` N ) = 0 )
51	50	adantr 481	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) /\ N e. NN0 ) -> ( (coeff ` F ) ` N ) = 0 )
52		eqidd 2623	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) /\ N e. NN0 ) -> ( (coeff ` G ) ` N ) = ( (coeff ` G ) ` N ) )
53	35, 39, 41, 41, 42, 51, 52	ofval 6906	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) /\ N e. NN0 ) -> ( ( (coeff ` F ) oF + (coeff ` G ) ) ` N ) = ( 0 + ( (coeff ` G ) ` N ) ) )
54	9, 53	mpdan 702	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( ( (coeff ` F ) oF + (coeff ` G ) ) ` N ) = ( 0 + ( (coeff ` G ) ` N ) ) )
55	37, 9	ffvelrnd 6360	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` G ) ` N ) e. CC )
56	55	addid2d 10237	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( 0 + ( (coeff ` G ) ` N ) ) = ( (coeff ` G ) ` N ) )
57	31, 54, 56	3eqtrd 2660	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` ( F oF + G ) ) ` N ) = ( (coeff ` G ) ` N ) )
58		simp2 1062	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> G e. (Poly ` S ) )
59		0red 10041	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> 0 e. RR )
60	14	nn0ge0d 11354	. . . . . . 7 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> 0 <_ M )
61	59, 15, 10, 60, 20	lelttrd 10195	. . . . . 6 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> 0 < N )
62	61	gt0ne0d 10592	. . . . 5 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N =/= 0 )
63	6, 28	dgreq0 24021	. . . . . . 7 |- ( G e. (Poly ` S ) -> ( G = 0p <-> ( (coeff ` G ) ` N ) = 0 ) )
64		fveq2 6191	. . . . . . . 8 |- ( G = 0p -> (deg ` G ) = (deg ` 0p ) )
65		dgr0 24018	. . . . . . . . 9 |- (deg ` 0p ) = 0
66	65	eqcomi 2631	. . . . . . . 8 |- 0 = (deg ` 0p )
67	64, 6, 66	3eqtr4g 2681	. . . . . . 7 |- ( G = 0p -> N = 0 )
68	63, 67	syl6bir 244	. . . . . 6 |- ( G e. (Poly ` S ) -> ( ( (coeff ` G ) ` N ) = 0 -> N = 0 ) )
69	68	necon3d 2815	. . . . 5 |- ( G e. (Poly ` S ) -> ( N =/= 0 -> ( (coeff ` G ) ` N ) =/= 0 ) )
70	58, 62, 69	sylc 65	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` G ) ` N ) =/= 0 )
71	57, 70	eqnetrd 2861	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> ( (coeff ` ( F oF + G ) ) ` N ) =/= 0 )
72		eqid 2622	. . . 4 |- (coeff ` ( F oF + G ) ) = (coeff ` ( F oF + G ) )
73		eqid 2622	. . . 4 |- (deg ` ( F oF + G ) ) = (deg ` ( F oF + G ) )
74	72, 73	dgrub 23990	. . 3 |- ( ( ( F oF + G ) e. (Poly ` CC ) /\ N e. NN0 /\ ( (coeff ` ( F oF + G ) ) ` N ) =/= 0 ) -> N <_ (deg ` ( F oF + G ) ) )
75	2, 9, 71, 74	syl3anc 1326	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> N <_ (deg ` ( F oF + G ) ) )
76	5, 10	letri3d 10179	. 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 ) ) ) ) )
77	26, 75, 76	mpbir2and 957	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) /\ M < N ) -> (deg ` ( F oF + G ) ) = N )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   ifcif 4086   class class class wbr 4653   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   + caddc 9939   < clt 10074   <_ cle 10075   NN0cn0 11292   0pc0p 23436  Polycply 23940  coeffccoe 23942  degcdgr 23943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-coe 23946  df-dgr 23947

This theorem is referenced by:  dgrcolem2  24030  plyremlem  24059