Theorem seqcoll2 12203

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 11488	. . . 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 6005	. . . . . . . 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 5631	. . . . . . 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 11780	. . . . . . . . . . . . 13 |- ( M ... N ) e. Fin
11		ssfi 7523	. . . . . . . . . . . . 13 |- ( ( ( M ... N ) e. Fin /\ A C_ ( M ... N ) ) -> A e. Fin )
12	10, 3, 11	sylancr 658	. . . . . . . . . . . 12 |- ( ph -> A e. Fin )
13		hasheq0 12117	. . . . . . . . . . . 12 |- ( A e. Fin -> ( ( # ` A ) = 0 <-> A = (/) ) )
14	12, 13	syl 16	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) = 0 <-> A = (/) ) )
15	14	necon3bbid 2634	. . . . . . . . . 10 |- ( ph -> ( -. ( # ` A ) = 0 <-> A =/= (/) ) )
16	9, 15	mpbird 232	. . . . . . . . 9 |- ( ph -> -. ( # ` A ) = 0 )
17		hashcl 12112	. . . . . . . . . . . 12 |- ( A e. Fin -> ( # ` A ) e. NN0 )
18	12, 17	syl 16	. . . . . . . . . . 11 |- ( ph -> ( # ` A ) e. NN0 )
19		elnn0 10571	. . . . . . . . . . 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 10886	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2525	. . . . . . 7 |- ( ph -> ( # ` A ) e. ( ZZ>= ` 1 ) )
25		eluzfz2 11448	. . . . . . 7 |- ( ( # ` A ) e. ( ZZ>= ` 1 ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
26	24, 25	syl 16	. . . . . 6 |- ( ph -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
27	8, 26	ffvelrnd 5834	. . . . 5 |- ( ph -> ( G ` ( # ` A ) ) e. A )
28	3, 27	sseldd 3347	. . . 4 |- ( ph -> ( G ` ( # ` A ) ) e. ( M ... N ) )
29	2, 28	sseldi 3344	. . 3 |- ( ph -> ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) )
30		elfzuz3 11439	. . . 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 11487	. . . . . . 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 3346	. . . . 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 467	. . . 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 11812	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) e. S )
39		peano2uz 10898	. . . . . . . 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 11486	. . . . . . 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 3346	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. ( M ... N ) )
44		eluzelre 10861	. . . . . . . . 9 |- ( ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) -> ( G ` ( # ` A ) ) e. RR )
45	29, 44	syl 16	. . . . . . . 8 |- ( ph -> ( G ` ( # ` A ) ) e. RR )
46	45	adantr 462	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) e. RR )
47		peano2re 9532	. . . . . . . 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 11442	. . . . . . . . 9 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. ZZ )
50	49	zred 10737	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. RR )
51	50	adantl 463	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. RR )
52	46	ltp1d 10253	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < ( ( G ` ( # ` A ) ) + 1 ) )
53		elfzle1 11443	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
54	53	adantl 463	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
55	46, 48, 51, 52, 54	ltletrd 9521	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < k )
56	6	adantr 462	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
57		f1ocnv 5643	. . . . . . . . . . . . 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 5631	. . . . . . . . . . . 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 751	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> k e. A )
62	60, 61	ffvelrnd 5834	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ( 1 ... ( # ` A ) ) )
63		elfzle2 11444	. . . . . . . . . 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 11442	. . . . . . . . . . . 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 10737	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. RR )
68	18	adantr 462	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. NN0 )
69	68	nn0red 10627	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. RR )
70	67, 69	lenltd 9510	. . . . . . . . 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 462	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
73	26	adantr 462	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
74		isorel 6006	. . . . . . . . . 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 1211	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) ) )
76		f1ocnvfv2 5973	. . . . . . . . . . 11 |- ( ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A /\ k e. A ) -> ( G ` ( `' G ` k ) ) = k )
77	56, 61, 76	syl2anc 656	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( G ` ( `' G ` k ) ) = k )
78	77	breq2d 4294	. . . . . . . . 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 612	. . . . . 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 3329	. . . 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 467	. . 3 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( F ` k ) = Z )
86	1, 29, 31, 38, 85	seqid2 11838	. 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 3358	. . 3 |- ( ph -> A C_ ( ZZ>= ` M ) )
90	33	ssdifd 3482	. . . . 5 |- ( ph -> ( ( M ... ( G ` ( # ` A ) ) ) \ A ) C_ ( ( M ... N ) \ A ) )
91	90	sselda 3346	. . . 4 |- ( ( ph /\ k e. ( ( M ... ( G ` ( # ` A ) ) ) \ A ) ) -> k e. ( ( M ... N ) \ A ) )
92	91, 84	syldan 467	. . 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 12202	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )
95	86, 94	eqtr3d 2469	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1364   e. wcel 1757   =/= wne 2598   \ cdif 3315   C_ wss 3318   (/)c0 3627   class class class wbr 4282   `'ccnv 4828   -->wf 5404   -1-1-onto->wf1o 5407   ` cfv 5408   Isom wiso 5409  (class class class)co 6082   Fincfn 7300   RRcr 9271   0cc0 9272   1c1 9273   + caddc 9275   < clt 9408   <_ cle 9409   NNcn 10312   NN0cn0 10569   ZZcz 10636   ZZ>=cuz 10851   ...cfz 11426   seqcseq 11792   #chash 12089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2416  ax-sep 4403  ax-nul 4411  ax-pow 4460  ax-pr 4521  ax-un 6363  ax-cnex 9328  ax-resscn 9329  ax-1cn 9330  ax-icn 9331  ax-addcl 9332  ax-addrcl 9333  ax-mulcl 9334  ax-mulrcl 9335  ax-mulcom 9336  ax-addass 9337  ax-mulass 9338  ax-distr 9339  ax-i2m1 9340  ax-1ne0 9341  ax-1rid 9342  ax-rnegex 9343  ax-rrecex 9344  ax-cnre 9345  ax-pre-lttri 9346  ax-pre-lttrn 9347  ax-pre-ltadd 9348  ax-pre-mulgt0 9349

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1702  df-eu 2260  df-mo 2261  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2966  df-sbc 3178  df-csb 3279  df-dif 3321  df-un 3323  df-in 3325  df-ss 3332  df-pss 3334  df-nul 3628  df-if 3782  df-pw 3852  df-sn 3868  df-pr 3870  df-tp 3872  df-op 3874  df-uni 4082  df-int 4119  df-iun 4163  df-br 4283  df-opab 4341  df-mpt 4342  df-tr 4376  df-eprel 4621  df-id 4625  df-po 4630  df-so 4631  df-fr 4668  df-we 4670  df-ord 4711  df-on 4712  df-lim 4713  df-suc 4714  df-xp 4835  df-rel 4836  df-cnv 4837  df-co 4838  df-dm 4839  df-rn 4840  df-res 4841  df-ima 4842  df-iota 5371  df-fun 5410  df-fn 5411  df-f 5412  df-f1 5413  df-fo 5414  df-f1o 5415  df-fv 5416  df-isom 5417  df-riota 6041  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-om 6468  df-1st 6568  df-2nd 6569  df-recs 6820  df-rdg 6854  df-1o 6910  df-er 7091  df-en 7301  df-dom 7302  df-sdom 7303  df-fin 7304  df-card 8099  df-pnf 9410  df-mnf 9411  df-xr 9412  df-ltxr 9413  df-le 9414  df-sub 9587  df-neg 9588  df-nn 10313  df-n0 10570  df-z 10637  df-uz 10852  df-fz 11427  df-seq 11793  df-hash 12090

This theorem is referenced by:  isercolllem3  13130  gsumval3OLD  16364  gsumval3  16367