Theorem coeaddlem 24005

Description: Lemma for coeadd 24007 and dgradd 24023. (Contributed by Mario Carneiro, 24-Jul-2014.)

Hypotheses

Ref	Expression
coefv0.1	|- A = (coeff ` F )
coeadd.2	|- B = (coeff ` G )
coeadd.3	|- M = (deg ` F )
coeadd.4	|- N = (deg ` G )

Assertion

Ref	Expression
coeaddlem	|- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF + G ) ) = ( A oF + B ) /\ (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) ) )

Proof of Theorem coeaddlem

Dummy variables k x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		plyaddcl 23976	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF + G ) e. (Poly ` CC ) )
2		coeadd.4	. . . . . 6 |- N = (deg ` G )
3		dgrcl 23989	. . . . . 6 |- ( G e. (Poly ` S ) -> (deg ` G ) e. NN0 )
4	2, 3	syl5eqel 2705	. . . . 5 |- ( G e. (Poly ` S ) -> N e. NN0 )
5	4	adantl 482	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> N e. NN0 )
6		coeadd.3	. . . . . 6 |- M = (deg ` F )
7		dgrcl 23989	. . . . . 6 |- ( F e. (Poly ` S ) -> (deg ` F ) e. NN0 )
8	6, 7	syl5eqel 2705	. . . . 5 |- ( F e. (Poly ` S ) -> M e. NN0 )
9	8	adantr 481	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> M e. NN0 )
10	5, 9	ifcld 4131	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> if ( M <_ N , N , M ) e. NN0 )
11		addcl 10018	. . . . 5 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
12	11	adantl 482	. . . 4 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
13		coefv0.1	. . . . . 6 |- A = (coeff ` F )
14	13	coef3 23988	. . . . 5 |- ( F e. (Poly ` S ) -> A : NN0 --> CC )
15	14	adantr 481	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> A : NN0 --> CC )
16		coeadd.2	. . . . . 6 |- B = (coeff ` G )
17	16	coef3 23988	. . . . 5 |- ( G e. (Poly ` S ) -> B : NN0 --> CC )
18	17	adantl 482	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> B : NN0 --> CC )
19		nn0ex 11298	. . . . 5 |- NN0 e. _V
20	19	a1i 11	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> NN0 e. _V )
21		inidm 3822	. . . 4 |- ( NN0 i^i NN0 ) = NN0
22	12, 15, 18, 20, 20, 21	off 6912	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( A oF + B ) : NN0 --> CC )
23		oveq12 6659	. . . . . . . . . 10 |- ( ( ( A ` k ) = 0 /\ ( B ` k ) = 0 ) -> ( ( A ` k ) + ( B ` k ) ) = ( 0 + 0 ) )
24		00id 10211	. . . . . . . . . 10 |- ( 0 + 0 ) = 0
25	23, 24	syl6eq 2672	. . . . . . . . 9 |- ( ( ( A ` k ) = 0 /\ ( B ` k ) = 0 ) -> ( ( A ` k ) + ( B ` k ) ) = 0 )
26		ffn 6045	. . . . . . . . . . . 12 |- ( A : NN0 --> CC -> A Fn NN0 )
27	15, 26	syl 17	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> A Fn NN0 )
28		ffn 6045	. . . . . . . . . . . 12 |- ( B : NN0 --> CC -> B Fn NN0 )
29	18, 28	syl 17	. . . . . . . . . . 11 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> B Fn NN0 )
30		eqidd 2623	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( A ` k ) = ( A ` k ) )
31		eqidd 2623	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( B ` k ) = ( B ` k ) )
32	27, 29, 20, 20, 21, 30, 31	ofval 6906	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( A oF + B ) ` k ) = ( ( A ` k ) + ( B ` k ) ) )
33	32	eqeq1d 2624	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( ( A oF + B ) ` k ) = 0 <-> ( ( A ` k ) + ( B ` k ) ) = 0 ) )
34	25, 33	syl5ibr 236	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( ( A ` k ) = 0 /\ ( B ` k ) = 0 ) -> ( ( A oF + B ) ` k ) = 0 ) )
35	34	necon3ad 2807	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( ( A oF + B ) ` k ) =/= 0 -> -. ( ( A ` k ) = 0 /\ ( B ` k ) = 0 ) ) )
36		neorian 2888	. . . . . . 7 |- ( ( ( A ` k ) =/= 0 \/ ( B ` k ) =/= 0 ) <-> -. ( ( A ` k ) = 0 /\ ( B ` k ) = 0 ) )
37	35, 36	syl6ibr 242	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( ( A oF + B ) ` k ) =/= 0 -> ( ( A ` k ) =/= 0 \/ ( B ` k ) =/= 0 ) ) )
38	13, 6	dgrub2 23991	. . . . . . . . . . 11 |- ( F e. (Poly ` S ) -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )
39	38	adantr 481	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } )
40		plyco0 23948	. . . . . . . . . . 11 |- ( ( M e. NN0 /\ A : NN0 --> CC ) -> ( ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ M ) ) )
41	9, 15, 40	syl2anc 693	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( A " ( ZZ>= ` ( M + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ M ) ) )
42	39, 41	mpbid 222	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> A. k e. NN0 ( ( A ` k ) =/= 0 -> k <_ M ) )
43	42	r19.21bi 2932	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( A ` k ) =/= 0 -> k <_ M ) )
44	9	adantr 481	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> M e. NN0 )
45	44	nn0red 11352	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> M e. RR )
46	5	adantr 481	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> N e. NN0 )
47	46	nn0red 11352	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> N e. RR )
48		max1 12016	. . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> M <_ if ( M <_ N , N , M ) )
49	45, 47, 48	syl2anc 693	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> M <_ if ( M <_ N , N , M ) )
50		nn0re 11301	. . . . . . . . . . 11 |- ( k e. NN0 -> k e. RR )
51	50	adantl 482	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> k e. RR )
52	10	adantr 481	. . . . . . . . . . 11 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> if ( M <_ N , N , M ) e. NN0 )
53	52	nn0red 11352	. . . . . . . . . 10 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> if ( M <_ N , N , M ) e. RR )
54		letr 10131	. . . . . . . . . 10 |- ( ( k e. RR /\ M e. RR /\ if ( M <_ N , N , M ) e. RR ) -> ( ( k <_ M /\ M <_ if ( M <_ N , N , M ) ) -> k <_ if ( M <_ N , N , M ) ) )
55	51, 45, 53, 54	syl3anc 1326	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( k <_ M /\ M <_ if ( M <_ N , N , M ) ) -> k <_ if ( M <_ N , N , M ) ) )
56	49, 55	mpan2d 710	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( k <_ M -> k <_ if ( M <_ N , N , M ) ) )
57	43, 56	syld 47	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( A ` k ) =/= 0 -> k <_ if ( M <_ N , N , M ) ) )
58	16, 2	dgrub2 23991	. . . . . . . . . . 11 |- ( G e. (Poly ` S ) -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )
59	58	adantl 482	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } )
60		plyco0 23948	. . . . . . . . . . 11 |- ( ( N e. NN0 /\ B : NN0 --> CC ) -> ( ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( B ` k ) =/= 0 -> k <_ N ) ) )
61	5, 18, 60	syl2anc 693	. . . . . . . . . 10 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( B " ( ZZ>= ` ( N + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( B ` k ) =/= 0 -> k <_ N ) ) )
62	59, 61	mpbid 222	. . . . . . . . 9 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> A. k e. NN0 ( ( B ` k ) =/= 0 -> k <_ N ) )
63	62	r19.21bi 2932	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( B ` k ) =/= 0 -> k <_ N ) )
64		max2 12018	. . . . . . . . . 10 |- ( ( M e. RR /\ N e. RR ) -> N <_ if ( M <_ N , N , M ) )
65	45, 47, 64	syl2anc 693	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> N <_ if ( M <_ N , N , M ) )
66		letr 10131	. . . . . . . . . 10 |- ( ( k e. RR /\ N e. RR /\ if ( M <_ N , N , M ) e. RR ) -> ( ( k <_ N /\ N <_ if ( M <_ N , N , M ) ) -> k <_ if ( M <_ N , N , M ) ) )
67	51, 47, 53, 66	syl3anc 1326	. . . . . . . . 9 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( k <_ N /\ N <_ if ( M <_ N , N , M ) ) -> k <_ if ( M <_ N , N , M ) ) )
68	65, 67	mpan2d 710	. . . . . . . 8 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( k <_ N -> k <_ if ( M <_ N , N , M ) ) )
69	63, 68	syld 47	. . . . . . 7 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( B ` k ) =/= 0 -> k <_ if ( M <_ N , N , M ) ) )
70	57, 69	jaod 395	. . . . . 6 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( ( A ` k ) =/= 0 \/ ( B ` k ) =/= 0 ) -> k <_ if ( M <_ N , N , M ) ) )
71	37, 70	syld 47	. . . . 5 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. NN0 ) -> ( ( ( A oF + B ) ` k ) =/= 0 -> k <_ if ( M <_ N , N , M ) ) )
72	71	ralrimiva 2966	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> A. k e. NN0 ( ( ( A oF + B ) ` k ) =/= 0 -> k <_ if ( M <_ N , N , M ) ) )
73		plyco0 23948	. . . . 5 |- ( ( if ( M <_ N , N , M ) e. NN0 /\ ( A oF + B ) : NN0 --> CC ) -> ( ( ( A oF + B ) " ( ZZ>= ` ( if ( M <_ N , N , M ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( ( A oF + B ) ` k ) =/= 0 -> k <_ if ( M <_ N , N , M ) ) ) )
74	10, 22, 73	syl2anc 693	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( ( A oF + B ) " ( ZZ>= ` ( if ( M <_ N , N , M ) + 1 ) ) ) = { 0 } <-> A. k e. NN0 ( ( ( A oF + B ) ` k ) =/= 0 -> k <_ if ( M <_ N , N , M ) ) ) )
75	72, 74	mpbird 247	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( ( A oF + B ) " ( ZZ>= ` ( if ( M <_ N , N , M ) + 1 ) ) ) = { 0 } )
76		simpl 473	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> F e. (Poly ` S ) )
77		simpr 477	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> G e. (Poly ` S ) )
78	13, 6	coeid 23994	. . . . 5 |- ( F e. (Poly ` S ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )
79	78	adantr 481	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> F = ( z e. CC |-> sum_ k e. ( 0 ... M ) ( ( A ` k ) x. ( z ^ k ) ) ) )
80	16, 2	coeid 23994	. . . . 5 |- ( G e. (Poly ` S ) -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )
81	80	adantl 482	. . . 4 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> G = ( z e. CC |-> sum_ k e. ( 0 ... N ) ( ( B ` k ) x. ( z ^ k ) ) ) )
82	76, 77, 9, 5, 15, 18, 39, 59, 79, 81	plyaddlem1 23969	. . 3 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( F oF + G ) = ( z e. CC |-> sum_ k e. ( 0 ... if ( M <_ N , N , M ) ) ( ( ( A oF + B ) ` k ) x. ( z ^ k ) ) ) )
83	1, 10, 22, 75, 82	coeeq 23983	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (coeff ` ( F oF + G ) ) = ( A oF + B ) )
84		elfznn0 12433	. . . 4 |- ( k e. ( 0 ... if ( M <_ N , N , M ) ) -> k e. NN0 )
85		ffvelrn 6357	. . . 4 |- ( ( ( A oF + B ) : NN0 --> CC /\ k e. NN0 ) -> ( ( A oF + B ) ` k ) e. CC )
86	22, 84, 85	syl2an 494	. . 3 |- ( ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) /\ k e. ( 0 ... if ( M <_ N , N , M ) ) ) -> ( ( A oF + B ) ` k ) e. CC )
87	1, 10, 86, 82	dgrle 23999	. 2 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) )
88	83, 87	jca 554	1 |- ( ( F e. (Poly ` S ) /\ G e. (Poly ` S ) ) -> ( (coeff ` ( F oF + G ) ) = ( A oF + B ) /\ (deg ` ( F oF + G ) ) <_ if ( M <_ N , N , M ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   <_ cle 10075   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   ^cexp 12860   sum_csu 14416  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:  coeadd  24007  dgradd  24023