Theorem seqcoll2 12628

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 11839	. . . 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 6216	. . . . . . . 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 5814	. . . . . . 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 12185	. . . . . . . . . . . . 13 |- ( M ... N ) e. Fin
11		ssfi 7792	. . . . . . . . . . . . 13 |- ( ( ( M ... N ) e. Fin /\ A C_ ( M ... N ) ) -> A e. Fin )
12	10, 3, 11	sylancr 669	. . . . . . . . . . . 12 |- ( ph -> A e. Fin )
13		hasheq0 12544	. . . . . . . . . . . 12 |- ( A e. Fin -> ( ( # ` A ) = 0 <-> A = (/) ) )
14	12, 13	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) = 0 <-> A = (/) ) )
15	14	necon3bbid 2661	. . . . . . . . . 10 |- ( ph -> ( -. ( # ` A ) = 0 <-> A =/= (/) ) )
16	9, 15	mpbird 236	. . . . . . . . 9 |- ( ph -> -. ( # ` A ) = 0 )
17		hashcl 12538	. . . . . . . . . . . 12 |- ( A e. Fin -> ( # ` A ) e. NN0 )
18	12, 17	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` A ) e. NN0 )
19		elnn0 10871	. . . . . . . . . . 11 |- ( ( # ` A ) e. NN0 <-> ( ( # ` A ) e. NN \/ ( # ` A ) = 0 ) )
20	18, 19	sylib 200	. . . . . . . . . 10 |- ( ph -> ( ( # ` A ) e. NN \/ ( # ` A ) = 0 ) )
21	20	ord 379	. . . . . . . . 9 |- ( ph -> ( -. ( # ` A ) e. NN -> ( # ` A ) = 0 ) )
22	16, 21	mt3d 129	. . . . . . . 8 |- ( ph -> ( # ` A ) e. NN )
23		nnuz 11194	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2539	. . . . . . 7 |- ( ph -> ( # ` A ) e. ( ZZ>= ` 1 ) )
25		eluzfz2 11807	. . . . . . 7 |- ( ( # ` A ) e. ( ZZ>= ` 1 ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
26	24, 25	syl 17	. . . . . 6 |- ( ph -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
27	8, 26	ffvelrnd 6023	. . . . 5 |- ( ph -> ( G ` ( # ` A ) ) e. A )
28	3, 27	sseldd 3433	. . . 4 |- ( ph -> ( G ` ( # ` A ) ) e. ( M ... N ) )
29	2, 28	sseldi 3430	. . 3 |- ( ph -> ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) )
30		elfzuz3 11797	. . . 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 11838	. . . . . . 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 3432	. . . . 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 473	. . . 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 12233	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) e. S )
39		peano2uz 11212	. . . . . . . 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 11837	. . . . . . 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 3432	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. ( M ... N ) )
44		eluzelre 11169	. . . . . . . . 9 |- ( ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) -> ( G ` ( # ` A ) ) e. RR )
45	29, 44	syl 17	. . . . . . . 8 |- ( ph -> ( G ` ( # ` A ) ) e. RR )
46	45	adantr 467	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) e. RR )
47		peano2re 9806	. . . . . . . 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 11800	. . . . . . . . 9 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. ZZ )
50	49	zred 11040	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. RR )
51	50	adantl 468	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. RR )
52	46	ltp1d 10537	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < ( ( G ` ( # ` A ) ) + 1 ) )
53		elfzle1 11802	. . . . . . . 8 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
54	53	adantl 468	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( ( G ` ( # ` A ) ) + 1 ) <_ k )
55	46, 48, 51, 52, 54	ltletrd 9795	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < k )
56	6	adantr 467	. . . . . . . . . . . . 13 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G : ( 1 ... ( # ` A ) ) -1-1-onto-> A )
57		f1ocnv 5826	. . . . . . . . . . . . 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 5814	. . . . . . . . . . . 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 766	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> k e. A )
62	60, 61	ffvelrnd 6023	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ( 1 ... ( # ` A ) ) )
63		elfzle2 11803	. . . . . . . . . 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 11800	. . . . . . . . . . . 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 11040	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. RR )
68	18	adantr 467	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. NN0 )
69	68	nn0red 10926	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. RR )
70	67, 69	lenltd 9781	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( `' G ` k ) <_ ( # ` A ) <-> -. ( # ` A ) < ( `' G ` k ) ) )
71	64, 70	mpbid 214	. . . . . . . 8 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> -. ( # ` A ) < ( `' G ` k ) )
72	4	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> G Isom < , < ( ( 1 ... ( # ` A ) ) , A ) )
73	26	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
74		isorel 6217	. . . . . . . . . 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 1266	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) ) )
76		f1ocnvfv2 6176	. . . . . . . . . . 11 |- ( ( G : ( 1 ... ( # ` A ) ) -1-1-onto-> A /\ k e. A ) -> ( G ` ( `' G ` k ) ) = k )
77	56, 61, 76	syl2anc 667	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( G ` ( `' G ` k ) ) = k )
78	77	breq2d 4414	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) <-> ( G ` ( # ` A ) ) < k ) )
79	75, 78	bitrd 257	. . . . . . . 8 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < k ) )
80	71, 79	mtbid 302	. . . . . . 7 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> -. ( G ` ( # ` A ) ) < k )
81	80	expr 620	. . . . . 6 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( k e. A -> -. ( G ` ( # ` A ) ) < k ) )
82	55, 81	mt2d 121	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> -. k e. A )
83	43, 82	eldifd 3415	. . . 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 473	. . 3 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( F ` k ) = Z )
86	1, 29, 31, 38, 85	seqid2 12259	. 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 3444	. . 3 |- ( ph -> A C_ ( ZZ>= ` M ) )
90	33	ssdifd 3569	. . . . 5 |- ( ph -> ( ( M ... ( G ` ( # ` A ) ) ) \ A ) C_ ( ( M ... N ) \ A ) )
91	90	sselda 3432	. . . 4 |- ( ( ph /\ k e. ( ( M ... ( G ` ( # ` A ) ) ) \ A ) ) -> k e. ( ( M ... N ) \ A ) )
92	91, 84	syldan 473	. . 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 12627	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )
95	86, 94	eqtr3d 2487	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   e. wcel 1887   =/= wne 2622   \ cdif 3401   C_ wss 3404   (/)c0 3731   class class class wbr 4402   `'ccnv 4833   -->wf 5578   -1-1-onto->wf1o 5581   ` cfv 5582   Isom wiso 5583  (class class class)co 6290   Fincfn 7569   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   seqcseq 12213   #chash 12515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-seq 12214  df-hash 12516

This theorem is referenced by:  isercolllem3  13730  gsumval3  17541