Theorem isercolllem3 13501

Description: Lemma for isercoll 13502. (Contributed by Mario Carneiro, 6-Apr-2015.)

Hypotheses

Ref	Expression
isercoll.z	|- Z = ( ZZ>= ` M )
isercoll.m	|- ( ph -> M e. ZZ )
isercoll.g	|- ( ph -> G : NN --> Z )
isercoll.i	|- ( ( ph /\ k e. NN ) -> ( G ` k ) < ( G ` ( k + 1 ) ) )
isercoll.0	|- ( ( ph /\ n e. ( Z \ ran G ) ) -> ( F ` n ) = 0 )
isercoll.f	|- ( ( ph /\ n e. Z ) -> ( F ` n ) e. CC )
isercoll.h	|- ( ( ph /\ k e. NN ) -> ( H ` k ) = ( F ` ( G ` k ) ) )

Assertion

Ref	Expression
isercolllem3	|- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( seq M ( + , F ) ` N ) = ( seq 1 ( + , H ) ` ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) )

Distinct variable groups:   k, n, F   k, N, n   ph, k, n   k, G, n   k, H, n   k, M, n   n, Z

Allowed substitution hint:   Z( k)

Proof of Theorem isercolllem3

Step	Hyp	Ref	Expression
1		addid2 9780	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 466	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		addid1 9777	. . 3 |- ( n e. CC -> ( n + 0 ) = n )
4	3	adantl 466	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. CC ) -> ( n + 0 ) = n )
5		addcl 9591	. . 3 |- ( ( n e. CC /\ k e. CC ) -> ( n + k ) e. CC )
6	5	adantl 466	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ ( n e. CC /\ k e. CC ) ) -> ( n + k ) e. CC )
7		0cnd 9606	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 0 e. CC )
8		cnvimass 5367	. . . . 5 |- ( `' G " ( M ... N ) ) C_ dom G
9		isercoll.g	. . . . . . 7 |- ( ph -> G : NN --> Z )
10	9	adantr 465	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> G : NN --> Z )
11		fdm 5741	. . . . . 6 |- ( G : NN --> Z -> dom G = NN )
12	10, 11	syl 16	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> dom G = NN )
13	8, 12	syl5sseq 3547	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( `' G " ( M ... N ) ) C_ NN )
14		isercoll.z	. . . . 5 |- Z = ( ZZ>= ` M )
15		isercoll.m	. . . . 5 |- ( ph -> M e. ZZ )
16		isercoll.i	. . . . 5 |- ( ( ph /\ k e. NN ) -> ( G ` k ) < ( G ` ( k + 1 ) ) )
17	14, 15, 9, 16	isercolllem1 13499	. . . 4 |- ( ( ph /\ ( `' G " ( M ... N ) ) C_ NN ) -> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( `' G " ( M ... N ) ) , ( G " ( `' G " ( M ... N ) ) ) ) )
18	13, 17	syldan 470	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( `' G " ( M ... N ) ) , ( G " ( `' G " ( M ... N ) ) ) ) )
19	14, 15, 9, 16	isercolllem2 13500	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) = ( `' G " ( M ... N ) ) )
20		isoeq4 6219	. . . 4 |- ( ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) = ( `' G " ( M ... N ) ) -> ( ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) , ( G " ( `' G " ( M ... N ) ) ) ) <-> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( `' G " ( M ... N ) ) , ( G " ( `' G " ( M ... N ) ) ) ) ) )
21	19, 20	syl 16	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) , ( G " ( `' G " ( M ... N ) ) ) ) <-> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( `' G " ( M ... N ) ) , ( G " ( `' G " ( M ... N ) ) ) ) ) )
22	18, 21	mpbird 232	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) , ( G " ( `' G " ( M ... N ) ) ) ) )
23	8	a1i 11	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( `' G " ( M ... N ) ) C_ dom G )
24		dfss1 3699	. . . . 5 |- ( ( `' G " ( M ... N ) ) C_ dom G <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = ( `' G " ( M ... N ) ) )
25	23, 24	sylib 196	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) = ( `' G " ( M ... N ) ) )
26		1nn 10567	. . . . . . 7 |- 1 e. NN
27	26	a1i 11	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. NN )
28		ffvelrn 6030	. . . . . . . . . 10 |- ( ( G : NN --> Z /\ 1 e. NN ) -> ( G ` 1 ) e. Z )
29	9, 26, 28	sylancl 662	. . . . . . . . 9 |- ( ph -> ( G ` 1 ) e. Z )
30	29, 14	syl6eleq 2555	. . . . . . . 8 |- ( ph -> ( G ` 1 ) e. ( ZZ>= ` M ) )
31	30	adantr 465	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G ` 1 ) e. ( ZZ>= ` M ) )
32		simpr 461	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> N e. ( ZZ>= ` ( G ` 1 ) ) )
33		elfzuzb 11707	. . . . . . 7 |- ( ( G ` 1 ) e. ( M ... N ) <-> ( ( G ` 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) )
34	31, 32, 33	sylanbrc 664	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G ` 1 ) e. ( M ... N ) )
35		ffn 5737	. . . . . . 7 |- ( G : NN --> Z -> G Fn NN )
36		elpreima 6008	. . . . . . 7 |- ( G Fn NN -> ( 1 e. ( `' G " ( M ... N ) ) <-> ( 1 e. NN /\ ( G ` 1 ) e. ( M ... N ) ) ) )
37	10, 35, 36	3syl 20	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( 1 e. ( `' G " ( M ... N ) ) <-> ( 1 e. NN /\ ( G ` 1 ) e. ( M ... N ) ) ) )
38	27, 34, 37	mpbir2and 922	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. ( `' G " ( M ... N ) ) )
39		ne0i 3799	. . . . 5 |- ( 1 e. ( `' G " ( M ... N ) ) -> ( `' G " ( M ... N ) ) =/= (/) )
40	38, 39	syl 16	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( `' G " ( M ... N ) ) =/= (/) )
41	25, 40	eqnetrd 2750	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
42		imadisj 5366	. . . 4 |- ( ( G " ( `' G " ( M ... N ) ) ) = (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = (/) )
43	42	necon3bii 2725	. . 3 |- ( ( G " ( `' G " ( M ... N ) ) ) =/= (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
44	41, 43	sylibr 212	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) =/= (/) )
45		ffun 5739	. . . 4 |- ( G : NN --> Z -> Fun G )
46		funimacnv 5666	. . . 4 |- ( Fun G -> ( G " ( `' G " ( M ... N ) ) ) = ( ( M ... N ) i^i ran G ) )
47	10, 45, 46	3syl 20	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) = ( ( M ... N ) i^i ran G ) )
48		inss1 3714	. . . 4 |- ( ( M ... N ) i^i ran G ) C_ ( M ... N )
49	48	a1i 11	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) i^i ran G ) C_ ( M ... N ) )
50	47, 49	eqsstrd 3533	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) C_ ( M ... N ) )
51		simpl 457	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ph )
52		elfzuz 11709	. . . 4 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
53	52, 14	syl6eleqr 2556	. . 3 |- ( n e. ( M ... N ) -> n e. Z )
54		isercoll.f	. . 3 |- ( ( ph /\ n e. Z ) -> ( F ` n ) e. CC )
55	51, 53, 54	syl2an 477	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( M ... N ) ) -> ( F ` n ) e. CC )
56	47	difeq2d 3618	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) )
57		difin 3742	. . . . . 6 |- ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) = ( ( M ... N ) \ ran G )
58	56, 57	syl6eq 2514	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ran G ) )
59	53	ssriv 3503	. . . . . 6 |- ( M ... N ) C_ Z
60		ssdif 3635	. . . . . 6 |- ( ( M ... N ) C_ Z -> ( ( M ... N ) \ ran G ) C_ ( Z \ ran G ) )
61	59, 60	mp1i 12	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ran G ) C_ ( Z \ ran G ) )
62	58, 61	eqsstrd 3533	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) C_ ( Z \ ran G ) )
63	62	sselda 3499	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) ) -> n e. ( Z \ ran G ) )
64		isercoll.0	. . . 4 |- ( ( ph /\ n e. ( Z \ ran G ) ) -> ( F ` n ) = 0 )
65	64	adantlr 714	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( Z \ ran G ) ) -> ( F ` n ) = 0 )
66	63, 65	syldan 470	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) ) -> ( F ` n ) = 0 )
67		elfznn 11739	. . . 4 |- ( k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) -> k e. NN )
68		isercoll.h	. . . 4 |- ( ( ph /\ k e. NN ) -> ( H ` k ) = ( F ` ( G ` k ) ) )
69	51, 67, 68	syl2an 477	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( H ` k ) = ( F ` ( G ` k ) ) )
70	19	eleq2d 2527	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) <-> k e. ( `' G " ( M ... N ) ) ) )
71	70	biimpa 484	. . . . 5 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> k e. ( `' G " ( M ... N ) ) )
72		fvres 5886	. . . . 5 |- ( k e. ( `' G " ( M ... N ) ) -> ( ( G |` ( `' G " ( M ... N ) ) ) ` k ) = ( G ` k ) )
73	71, 72	syl 16	. . . 4 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( ( G |` ( `' G " ( M ... N ) ) ) ` k ) = ( G ` k ) )
74	73	fveq2d 5876	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( F ` ( ( G |` ( `' G " ( M ... N ) ) ) ` k ) ) = ( F ` ( G ` k ) ) )
75	69, 74	eqtr4d 2501	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( H ` k ) = ( F ` ( ( G |` ( `' G " ( M ... N ) ) ) ` k ) ) )
76	2, 4, 6, 7, 22, 44, 50, 55, 66, 75	seqcoll2 12517	1 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( seq M ( + , F ) ` N ) = ( seq 1 ( + , H ) ` ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   e. wcel 1819   =/= wne 2652   \ cdif 3468   i^i cin 3470   C_ wss 3471   (/)c0 3793   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   ran crn 5009   |` cres 5010   "cima 5011   Fun wfun 5588   Fn wfn 5589   -->wf 5590   ` cfv 5594   Isom wiso 5595  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510   + caddc 9512   < clt 9645   NNcn 10556   ZZcz 10885   ZZ>=cuz 11106   ...cfz 11697   seqcseq 12110   #chash 12408

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-rep 4568  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  ax-pre-sup 9587

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-rmo 2815  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-sup 7919  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 12111  df-hash 12409

This theorem is referenced by:  isercoll  13502