Theorem seqcoll2 12516

Description: The function F contains a sparse set of non-zero values to be summed. The function G is an order isomorphism from the set of non-zero values of F to a 1-based finite sequence, and H collects these non-zero values together. Under these conditions, the sum over the values in H yields the same result as the sum over the original set F. (Contributed by Mario Carneiro, 13-Dec-2014.)

Hypotheses

Ref	Expression
seqcoll2.1	|- ( ( ph /\ k e. S ) -> ( Z .+ k ) = k )
seqcoll2.1b	|- ( ( ph /\ k e. S ) -> ( k .+ Z ) = k )
seqcoll2.c	|- ( ( ph /\ ( k e. S /\ n e. S ) ) -> ( k .+ n ) e. S )
seqcoll2.a	|- ( ph -> Z e. S )
seqcoll2.2	|- ( ph -> G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
seqcoll2.3	|- ( ph -> A =/= (/) )
seqcoll2.5	|- ( ph -> A C_ ( M ... N ) )
seqcoll2.6	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )
seqcoll2.7	|- ( ( ph /\ k e. ( ( M ... N ) \ A ) ) -> ( F ` k ) = Z )
seqcoll2.8	|- ( ( ph /\ n e. ( 1 ... ( # ` A ) ) ) -> ( H ` n ) = ( F ` ( G ` n ) ) )

Assertion

Ref	Expression
seqcoll2	|- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )

Distinct variable groups:   k, n, A   k, F, n   k, G, n   n, H   k, M, n   ph, k, n   k, N   .+ , k, n   S, k, n   k, Z

Allowed substitution hints:   H( k)   N( n)   Z( n)

Proof of Theorem seqcoll2

Step	Hyp	Ref	Expression
1		seqcoll2.1b	. . 3 |- ( ( ph /\ k e. S ) -> ( k .+ Z ) = k )
2		fzssuz 11749	. . . 4 |- ( M ... N ) C_ ( ZZ>= ` M )
3		seqcoll2.5	. . . . 5 |- ( ph -> A C_ ( M ... N ) )
4		seqcoll2.2	. . . . . . . 8 |- ( ph -> G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
5		isof1o 6222	. . . . . . . 8 |- ( G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
6	4, 5	syl 16	. . . . . . 7 |- ( ph -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
7		f1of 5822	. . . . . . 7 |- ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> G : ( 1 ... ( # ` A ) ) --> A )
8	6, 7	syl 16	. . . . . 6 |- ( ph -> G : ( 1 ... ( # ` A ) ) --> A )
9		seqcoll2.3	. . . . . . . . . 10 |- ( ph -> A =/= (/) )
10		fzfi 12084	. . . . . . . . . . . . 13 |- ( M ... N ) e. Fin
11		ssfi 7759	. . . . . . . . . . . . 13 |- ( ( ( M ... N ) e. Fin /\ A C_ ( M ... N ) ) -> A e. Fin )
12	10, 3, 11	sylancr 663	. . . . . . . . . . . 12 |- ( ph -> A e. Fin )
13		hasheq0 12435	. . . . . . . . . . . 12 |- ( A e. Fin -> ( ( # ` A ) = 0 <-> A = (/) ) )
14	12, 13	syl 16	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) = 0 <-> A = (/) ) )
15	14	necon3bbid 2704	. . . . . . . . . 10 |- ( ph -> ( -. ( # ` A ) = 0 <-> A =/= (/) ) )
16	9, 15	mpbird 232	. . . . . . . . 9 |- ( ph -> -. ( # ` A ) = 0 )
17		hashcl 12430	. . . . . . . . . . . 12 |- ( A e. Fin -> ( # ` A ) e. NN0 )
18	12, 17	syl 16	. . . . . . . . . . 11 |- ( ph -> ( # ` A ) e. NN0 )
19		elnn0 10818	. . . . . . . . . . 11 |- ( ( # ` A ) e. NN0 <-> ( ( # ` A ) e. NN \/ ( # ` A ) = 0 ) )
20	18, 19	sylib 196	. . . . . . . . . 10 |- ( ph -> ( ( # ` A ) e. NN \/ ( # ` A ) = 0 ) )
21	20	ord 377	. . . . . . . . 9 |- ( ph -> ( -. ( # ` A ) e. NN -> ( # ` A ) = 0 ) )
22	16, 21	mt3d 125	. . . . . . . 8 |- ( ph -> ( # ` A ) e. NN )
23		nnuz 11141	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2555	. . . . . . 7 |- ( ph -> ( # ` A ) e. ( ZZ>= ` 1 ) )
25		eluzfz2 11719	. . . . . . 7 |- ( ( # ` A ) e. ( ZZ>= ` 1 ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
26	24, 25	syl 16	. . . . . 6 |- ( ph -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
27	8, 26	ffvelrnd 6033	. . . . 5 |- ( ph -> ( G ` ( # ` A ) ) e. A )
28	3, 27	sseldd 3500	. . . 4 |- ( ph -> ( G ` ( # ` A ) ) e. ( M ... N ) )
29	2, 28	sseldi 3497	. . 3 |- ( ph -> ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) )
30		elfzuz3 11710	. . . 4 |- ( ( G ` ( # ` A ) ) e. ( M ... N ) -> N e. ( ZZ>= ` ( G ` ( # ` A ) ) ) )
31	28, 30	syl 16	. . 3 |- ( ph -> N e. ( ZZ>= ` ( G ` ( # ` A ) ) ) )
32		fzss2 11748	. . . . . . 7 |- ( N e. ( ZZ>= ` ( G ` ( # ` A ) ) ) -> ( M ... ( G ` ( # ` A ) ) ) C_ ( M ... N ) )
33	31, 32	syl 16	. . . . . 6 |- ( ph -> ( M ... ( G ` ( # ` A ) ) ) C_ ( M ... N ) )
34	33	sselda 3499	. . . . 5 |- ( ( ph /\ k e. ( M ... ( G ` ( # ` A ) ) ) ) -> k e. ( M ... N ) )
35		seqcoll2.6	. . . . 5 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. S )
36	34, 35	syldan 470	. . . 4 |- ( ( ph /\ k e. ( M ... ( G ` ( # ` A ) ) ) ) -> ( F ` k ) e. S )
37		seqcoll2.c	. . . 4 |- ( ( ph /\ ( k e. S /\ n e. S ) ) -> ( k .+ n ) e. S )
38	29, 36, 37	seqcl 12129	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) e. S )
39		peano2uz 11159	. . . . . . . 8 |- ( ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) -> ( ( G ` ( # ` A ) ) + 1 ) e. ( ZZ>= ` M ) )
40	29, 39	syl 16	. . . . . . 7 |- ( ph -> ( ( G ` ( # ` A ) ) + 1 ) e. ( ZZ>= ` M ) )
41		fzss1 11747	. . . . . . 7 |- ( ( ( G ` ( # ` A ) ) + 1 ) e. ( ZZ>= ` M ) -> ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) C_ ( M ... N ) )
42	40, 41	syl 16	. . . . . 6 |- ( ph -> ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) C_ ( M ... N ) )
43	42	sselda 3499	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. ( M ... N ) )
44		eluzelre 11116	. . . . . . . . 9 |- ( ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) -> ( G ` ( # ` A ) ) e. RR )
45	29, 44	syl 16	. . . . . . . 8 |- ( ph -> ( G ` ( # ` A ) ) e. RR )
46	45	adantr 465	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) e. RR )
47		peano2re 9770	. . . . . . . 8 |- ( ( G ` ( # ` A ) ) e. RR -> ( ( G ` ( # ` A ) ) + 1 ) e. RR )
48	46, 47	syl 16	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( ( G ` ( # ` A ) ) + 1 ) e. RR )
49		elfzelz 11713	. . . . . . . . 9 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. ZZ )
50	49	zred 10990	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. RR )
51	50	adantl 466	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. RR )
52	46	ltp1d 10496	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < ( ( G ` ( # ` A ) ) + 1 ) )
53		elfzle1 11714	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
54	53	adantl 466	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
55	46, 48, 51, 52, 54	ltletrd 9759	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < k )
56	6	adantr 465	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
57		f1ocnv 5834	. . . . . . . . . . . . 13 |- ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> `' G : A -1-1-onto-> ( 1 ... ( # ` A ) ) )
58	56, 57	syl 16	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> `' G : A -1-1-onto-> ( 1 ... ( # ` A ) ) )
59		f1of 5822	. . . . . . . . . . . 12 |- ( `' G : A -1-1-onto-> ( 1 ... ( # ` A ) ) -> `' G : A --> ( 1 ... ( # ` A ) ) )
60	58, 59	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> `' G : A --> ( 1 ... ( # ` A ) ) )
61		simprr 757	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> k e. A )
62	60, 61	ffvelrnd 6033	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ( 1 ... ( # ` A ) ) )
63		elfzle2 11715	. . . . . . . . . 10 |- ( ( `' G ` k ) e. ( 1 ... ( # ` A ) ) -> ( `' G ` k ) <_ ( # ` A ) )
64	62, 63	syl 16	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) <_ ( # ` A ) )
65		elfzelz 11713	. . . . . . . . . . . 12 |- ( ( `' G ` k ) e. ( 1 ... ( # ` A ) ) -> ( `' G ` k ) e. ZZ )
66	62, 65	syl 16	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ZZ )
67	66	zred 10990	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. RR )
68	18	adantr 465	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. NN0 )
69	68	nn0red 10874	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. RR )
70	67, 69	lenltd 9748	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( `' G ` k ) <_ ( # ` A ) <-> -. ( # ` A ) < ( `' G ` k ) ) )
71	64, 70	mpbid 210	. . . . . . . 8 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> -. ( # ` A ) < ( `' G ` k ) )
72	4	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
73	26	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
74		isorel 6223	. . . . . . . . . 10 |- ( ( G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) /\ ( ( # ` A ) e. ( 1 ... ( # ` A ) ) /\ ( `' G ` k ) e. ( 1 ... ( # ` A ) ) ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) ) )
75	72, 73, 62, 74	syl12anc 1226	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) ) )
76		f1ocnvfv2 6184	. . . . . . . . . . 11 |- ( ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A /\ k e. A ) -> ( G ` ( `' G ` k ) ) = k )
77	56, 61, 76	syl2anc 661	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( G ` ( `' G ` k ) ) = k )
78	77	breq2d 4468	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) <-> ( G ` ( # ` A ) ) < k ) )
79	75, 78	bitrd 253	. . . . . . . 8 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < k ) )
80	71, 79	mtbid 300	. . . . . . 7 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> -. ( G ` ( # ` A ) ) < k )
81	80	expr 615	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( k e. A -> -. ( G ` ( # ` A ) ) < k ) )
82	55, 81	mt2d 117	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> -. k e. A )
83	43, 82	eldifd 3482	. . . 4 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. ( ( M ... N ) \ A ) )
84		seqcoll2.7	. . . 4 |- ( ( ph /\ k e. ( ( M ... N ) \ A ) ) -> ( F ` k ) = Z )
85	83, 84	syldan 470	. . 3 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( F ` k ) = Z )
86	1, 29, 31, 38, 85	seqid2 12155	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) = ( seq M ( .+ , F ) ` N ) )
87		seqcoll2.1	. . 3 |- ( ( ph /\ k e. S ) -> ( Z .+ k ) = k )
88		seqcoll2.a	. . 3 |- ( ph -> Z e. S )
89	3, 2	syl6ss 3511	. . 3 |- ( ph -> A C_ ( ZZ>= ` M ) )
90	33	ssdifd 3636	. . . . 5 |- ( ph -> ( ( M ... ( G ` ( # ` A ) ) ) \ A ) C_ ( ( M ... N ) \ A ) )
91	90	sselda 3499	. . . 4 |- ( ( ph /\ k e. ( ( M ... ( G ` ( # ` A ) ) ) \ A ) ) -> k e. ( ( M ... N ) \ A ) )
92	91, 84	syldan 470	. . 3 |- ( ( ph /\ k e. ( ( M ... ( G ` ( # ` A ) ) ) \ A ) ) -> ( F ` k ) = Z )
93		seqcoll2.8	. . 3 |- ( ( ph /\ n e. ( 1 ... ( # ` A ) ) ) -> ( H ` n ) = ( F ` ( G ` n ) ) )
94	87, 1, 37, 88, 4, 26, 89, 36, 92, 93	seqcoll 12515	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )
95	86, 94	eqtr3d 2500	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   \ cdif 3468   C_ wss 3471   (/)c0 3793   class class class wbr 4456   `'ccnv 5007   -->wf 5590   -1-1-onto->wf1o 5593   ` cfv 5594   Isom wiso 5595  (class class class)co 6296   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510   + caddc 9512   < clt 9645   <_ cle 9646   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697   seqcseq 12109   #chash 12407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12110  df-hash 12408

This theorem is referenced by:  isercolllem3  13500  gsumval3OLD  17034  gsumval3  17037