Theorem seqcoll2 12217

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 11499	. . . 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 6016	. . . . . . . 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 5641	. . . . . . 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 11794	. . . . . . . . . . . . 13 |- ( M ... N ) e. Fin
11		ssfi 7533	. . . . . . . . . . . . 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 12131	. . . . . . . . . . . 12 |- ( A e. Fin -> ( ( # ` A ) = 0 <-> A = (/) ) )
14	12, 13	syl 16	. . . . . . . . . . 11 |- ( ph -> ( ( # ` A ) = 0 <-> A = (/) ) )
15	14	necon3bbid 2642	. . . . . . . . . 10 |- ( ph -> ( -. ( # ` A ) = 0 <-> A =/= (/) ) )
16	9, 15	mpbird 232	. . . . . . . . 9 |- ( ph -> -. ( # ` A ) = 0 )
17		hashcl 12126	. . . . . . . . . . . 12 |- ( A e. Fin -> ( # ` A ) e. NN0 )
18	12, 17	syl 16	. . . . . . . . . . 11 |- ( ph -> ( # ` A ) e. NN0 )
19		elnn0 10581	. . . . . . . . . . 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 10896	. . . . . . . 8 |- NN = ( ZZ>= ` 1 )
24	22, 23	syl6eleq 2533	. . . . . . 7 |- ( ph -> ( # ` A ) e. ( ZZ>= ` 1 ) )
25		eluzfz2 11459	. . . . . . 7 |- ( ( # ` A ) e. ( ZZ>= ` 1 ) -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
26	24, 25	syl 16	. . . . . 6 |- ( ph -> ( # ` A ) e. ( 1 ... ( # ` A ) ) )
27	8, 26	ffvelrnd 5844	. . . . 5 |- ( ph -> ( G ` ( # ` A ) ) e. A )
28	3, 27	sseldd 3357	. . . 4 |- ( ph -> ( G ` ( # ` A ) ) e. ( M ... N ) )
29	2, 28	sseldi 3354	. . 3 |- ( ph -> ( G ` ( # ` A ) ) e. ( ZZ>= ` M ) )
30		elfzuz3 11450	. . . 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 11498	. . . . . . 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 3356	. . . . 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 11826	. . 3 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) e. S )
39		peano2uz 10908	. . . . . . . 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 11497	. . . . . . 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 3356	. . . . 5 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> k e. ( M ... N ) )
44		eluzelre 10871	. . . . . . . . 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 9542	. . . . . . . 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 11453	. . . . . . . . 9 |- ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) -> k e. ZZ )
50	49	zred 10747	. . . . . . . 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 10263	. . . . . . 7 |- ( ( ph /\ k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) ) -> ( G ` ( # ` A ) ) < ( ( G ` ( # ` A ) ) + 1 ) )
53		elfzle1 11454	. . . . . . . 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 9531	. . . . . 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 5653	. . . . . . . . . . . . 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 5641	. . . . . . . . . . . 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 756	. . . . . . . . . . 11 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> k e. A )
62	60, 61	ffvelrnd 5844	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( `' G ` k ) e. ( 1 ... ( # ` A ) ) )
63		elfzle2 11455	. . . . . . . . . 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 11453	. . . . . . . . . . . 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 10747	. . . . . . . . . 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 10637	. . . . . . . . . 10 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( # ` A ) e. RR )
70	67, 69	lenltd 9520	. . . . . . . . 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 6017	. . . . . . . . . 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 1216	. . . . . . . . 9 |- ( ( ph /\ ( k e. ( ( ( G ` ( # ` A ) ) + 1 ) ... N ) /\ k e. A ) ) -> ( ( # ` A ) < ( `' G ` k ) <-> ( G ` ( # ` A ) ) < ( G ` ( `' G ` k ) ) ) )
76		f1ocnvfv2 5984	. . . . . . . . . . 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 4304	. . . . . . . . 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 3339	. . . 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 11852	. 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 3368	. . 3 |- ( ph -> A C_ ( ZZ>= ` M ) )
90	33	ssdifd 3492	. . . . 5 |- ( ph -> ( ( M ... ( G ` ( # ` A ) ) ) \ A ) C_ ( ( M ... N ) \ A ) )
91	90	sselda 3356	. . . 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 12216	. 2 |- ( ph -> ( seq M ( .+ , F ) ` ( G ` ( # ` A ) ) ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )
95	86, 94	eqtr3d 2477	1 |- ( ph -> ( seq M ( .+ , F ) ` N ) = ( seq 1 ( .+ , H ) ` ( # ` A ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   \/ wo 368   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2606   \ cdif 3325   C_ wss 3328   (/)c0 3637   class class class wbr 4292   `'ccnv 4839   -->wf 5414   -1-1-onto->wf1o 5417   ` cfv 5418   Isom wiso 5419  (class class class)co 6091   Fincfn 7310   RRcr 9281   0cc0 9282   1c1 9283   + caddc 9285   < clt 9418   <_ cle 9419   NNcn 10322   NN0cn0 10579   ZZcz 10646   ZZ>=cuz 10861   ...cfz 11437   seqcseq 11806   #chash 12103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-n0 10580  df-z 10647  df-uz 10862  df-fz 11438  df-seq 11807  df-hash 12104

This theorem is referenced by:  isercolllem3  13144  gsumval3OLD  16382  gsumval3  16385