Theorem seqcoll2 13249

Description: The function F contains a sparse set of nonzero values to be summed. The function G is an order isomorphism from the set of nonzero values of F to a 1-based finite sequence, and H collects these nonzero 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 12382	. . . 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 6573	. . . . . . . 8 |- ( G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
6	4, 5	syl 17	. . . . . . 7 |- ( ph -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
7		f1of 6137	. . . . . . 7 |- ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> G : ( 1 ... ( # ` A ) ) --> A )
8	6, 7	syl 17	. . . . . 6 |- ( ph -> G : ( 1 ... ( # ` A ) ) --> A )
9		seqcoll2.3	. . . . . . . . . 10 |- ( ph -> A =/= (/) )
10		fzfi 12771	. . . . . . . . . . . . 13 |- ( M ... N ) e. Fin
11		ssfi 8180	. . . . . . . . . . . . 13 |- ( ( ( M ... N ) e. Fin /\ A C_ ( M ... N ) ) -> A e. Fin )
12	10, 3, 11	sylancr 695	. . . . . . . . . . . 12 |- ( ph -> A e. Fin )
13		hasheq0 13154	. . . . . . . . . . . 12 |- ( A e. Fin -> ( ( # ` A ) = 0 <-> A = (/) ) )
14	12, 13	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) = 0 <-> A = (/) ) )
15	14	necon3bbid 2831	. . . . . . . . . 10 |- ( ph -> ( -. ( # ` A ) = 0 <-> A =/= (/) ) )
16	9, 15	mpbird 247	. . . . . . . . 9 |- ( ph -> -. ( # ` A ) = 0 )
17		hashcl 13147	. . . . . . . . . . . 12 |- ( A e. Fin -> ( # ` A ) e. NN0 )
18	12, 17	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` A ) e. NN0 )
19		elnn0 11294	. . . . . . . . . . 11 |- ( ( # ` A ) e. NN0 <-> ( ( # ` A ) e. NN \/ ( # ` A ) = 0 ) )
20	18, 19	sylib 208	. . . . . . . . . 10 |- ( ph -> ( ( # ` A ) e. NN \/ ( # ` A ) = 0 ) )
21	20	ord 392	. . . . . . . . 9 |- ( ph -> ( -. ( # ` A ) e. NN -> ( # ` A ) = 0 ) )
22	16, 21	mt3d 140	. . . . . . . 8 |- ( ph -> ( # ` A ) e. NN )
23		nnuz 11723	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2711	. . . . . . 7 |- ( ph -> ( # ` A ) e. ( ZZ>= ` 1 ) )
25		eluzfz2 12349	. . . . . . 7 |- ( ( # ` A ) e. ( ZZ>= ` 1 ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
26	24, 25	syl 17	. . . . . 6 |- ( ph -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
27	8, 26	ffvelrnd 6360	. . . . 5 |- ( ph -> ( G ` ( # ` A ) ) e. A )
28	3, 27	sseldd 3604	. . . 4 |- ( ph -> ( G ` ( # ` A ) ) e. ( M ... N ) )
29	2, 28	sseldi 3601	. . 3 |- ( ph -> ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) )
30		elfzuz3 12339	. . . 4 |- ( ( G ` ( # ` A ) ) e. ( M ... N ) -> N e. ( ZZ>= ` ( G ` ( # ` A ) ) ) )
31	28, 30	syl 17	. . 3 |- ( ph -> N e. ( ZZ>= ` ( G ` ( # ` A ) ) ) )
32		fzss2 12381	. . . . . . 7 |- ( N e. ( ZZ>= ` ( G ` ( # ` A ) ) ) -> ( M ... ( G ` ( # ` A ) ) ) C_ ( M ... N ) )
33	31, 32	syl 17	. . . . . 6 |- ( ph -> ( M ... ( G ` ( # ` A ) ) ) C_ ( M ... N ) )
34	33	sselda 3603	. . . . 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 487	. . . 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 12821	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) e. S )
39		peano2uz 11741	. . . . . . . 8 |- ( ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) -> ( ( G ` ( # ` A ) ) + 1 ) e. ( ZZ>= ` M ) )
40	29, 39	syl 17	. . . . . . 7 |- ( ph -> ( ( G ` ( # ` A ) ) + 1 ) e. ( ZZ>= ` M ) )
41		fzss1 12380	. . . . . . 7 |- ( ( ( G ` ( # ` A ) ) + 1 ) e. ( ZZ>= ` M ) -> ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) C_ ( M ... N ) )
42	40, 41	syl 17	. . . . . 6 |- ( ph -> ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) C_ ( M ... N ) )
43	42	sselda 3603	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. ( M ... N ) )
44		eluzelre 11698	. . . . . . . . 9 |- ( ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) -> ( G ` ( # ` A ) ) e. RR )
45	29, 44	syl 17	. . . . . . . 8 |- ( ph -> ( G ` ( # ` A ) ) e. RR )
46	45	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) e. RR )
47		peano2re 10209	. . . . . . . 8 |- ( ( G ` ( # ` A ) ) e. RR -> ( ( G ` ( # ` A ) ) + 1 ) e. RR )
48	46, 47	syl 17	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( ( G ` ( # ` A ) ) + 1 ) e. RR )
49		elfzelz 12342	. . . . . . . . 9 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. ZZ )
50	49	zred 11482	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. RR )
51	50	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. RR )
52	46	ltp1d 10954	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < ( ( G ` ( # ` A ) ) + 1 ) )
53		elfzle1 12344	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
54	53	adantl 482	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
55	46, 48, 51, 52, 54	ltletrd 10197	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < k )
56	6	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
57		f1ocnv 6149	. . . . . . . . . . . . 13 |- ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A -> `' G : A -1-1-onto-> ( 1 ... ( # ` A ) ) )
58	56, 57	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> `' G : A -1-1-onto-> ( 1 ... ( # ` A ) ) )
59		f1of 6137	. . . . . . . . . . . 12 |- ( `' G : A -1-1-onto-> ( 1 ... ( # ` A ) ) -> `' G : A --> ( 1 ... ( # ` A ) ) )
60	58, 59	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> `' G : A --> ( 1 ... ( # ` A ) ) )
61		simprr 796	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> k e. A )
62	60, 61	ffvelrnd 6360	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ( 1 ... ( # ` A ) ) )
63		elfzle2 12345	. . . . . . . . . 10 |- ( ( `' G ` k ) e. ( 1 ... ( # ` A ) ) -> ( `' G ` k ) <_ ( # ` A ) )
64	62, 63	syl 17	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) <_ ( # ` A ) )
65		elfzelz 12342	. . . . . . . . . . . 12 |- ( ( `' G ` k ) e. ( 1 ... ( # ` A ) ) -> ( `' G ` k ) e. ZZ )
66	62, 65	syl 17	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ZZ )
67	66	zred 11482	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. RR )
68	18	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. NN0 )
69	68	nn0red 11352	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. RR )
70	67, 69	lenltd 10183	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( `' G ` k ) <_ ( # ` A ) <-> -. ( # ` A ) < ( `' G ` k ) ) )
71	64, 70	mpbid 222	. . . . . . . 8 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> -. ( # ` A ) < ( `' G ` k ) )
72	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
73	26	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
74		isorel 6576	. . . . . . . . . 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 1324	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) ) )
76		f1ocnvfv2 6533	. . . . . . . . . . 11 |- ( ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A /\ k e. A ) -> ( G ` ( `' G ` k ) ) = k )
77	56, 61, 76	syl2anc 693	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( G ` ( `' G ` k ) ) = k )
78	77	breq2d 4665	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) <-> ( G ` ( # ` A ) ) < k ) )
79	75, 78	bitrd 268	. . . . . . . 8 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < k ) )
80	71, 79	mtbid 314	. . . . . . 7 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> -. ( G ` ( # ` A ) ) < k )
81	80	expr 643	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( k e. A -> -. ( G ` ( # ` A ) ) < k ) )
82	55, 81	mt2d 131	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> -. k e. A )
83	43, 82	eldifd 3585	. . . 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 487	. . 3 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( F ` k ) = Z )
86	1, 29, 31, 38, 85	seqid2 12847	. 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 3615	. . 3 |- ( ph -> A C_ ( ZZ>= ` M ) )
90	33	ssdifd 3746	. . . . 5 |- ( ph -> ( ( M ... ( G ` ( # ` A ) ) ) \ A ) C_ ( ( M ... N ) \ A ) )
91	90	sselda 3603	. . . 4 |- ( ( ph /\ k e. ( ( M ... ( G ` ( # ` A ) ) ) \ A ) ) -> k e. ( ( M ... N ) \ A ) )
92	91, 84	syldan 487	. . 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 13248	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )
95	86, 94	eqtr3d 2658	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   C_ wss 3574   (/)c0 3915   class class class wbr 4653   `'ccnv 5113   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-hash 13118

This theorem is referenced by:  isercolllem3  14397  gsumval3  18308