Theorem isercolllem3 13156

Description: Lemma for isercoll 13157. (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 9564	. . 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 9561	. . 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 9376	. . 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 9391	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 0 e. CC )
8		cnvimass 5201	. . . . 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 5575	. . . . . 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 3416	. . . 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 13154	. . . 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 13155	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) = ( `' G " ( M ... N ) ) )
20		isoeq4 6025	. . . 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 3567	. . . . 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 10345	. . . . . . 7 |- 1 e. NN
27	26	a1i 11	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. NN )
28		ffvelrn 5853	. . . . . . . . . 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 2533	. . . . . . . 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 11459	. . . . . . 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 5571	. . . . . . 7 |- ( G : NN --> Z -> G Fn NN )
36		elpreima 5835	. . . . . . 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 913	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. ( `' G " ( M ... N ) ) )
39		ne0i 3655	. . . . 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 2638	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
42		imadisj 5200	. . . 4 |- ( ( G " ( `' G " ( M ... N ) ) ) = (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = (/) )
43	42	necon3bii 2652	. . 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 5573	. . . 4 |- ( G : NN --> Z -> Fun G )
46		funimacnv 5502	. . . 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 3582	. . . 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 3402	. 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 11461	. . . 4 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
53	52, 14	syl6eleqr 2534	. . 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 3486	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) )
57		difin 3599	. . . . . 6 |- ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) = ( ( M ... N ) \ ran G )
58	56, 57	syl6eq 2491	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ran G ) )
59	53	ssriv 3372	. . . . . 6 |- ( M ... N ) C_ Z
60		ssdif 3503	. . . . . 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 3402	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) C_ ( Z \ ran G ) )
63	62	sselda 3368	. . 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 11490	. . . 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 2510	. . . . . 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 5716	. . . . 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 5707	. . 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 2478	. 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 12229	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 1369   e. wcel 1756   =/= wne 2618   \ cdif 3337   i^i cin 3339   C_ wss 3340   (/)c0 3649   class class class wbr 4304   `'ccnv 4851   dom cdm 4852   ran crn 4853   |` cres 4854   "cima 4855   Fun wfun 5424   Fn wfn 5425   -->wf 5426   ` cfv 5430   Isom wiso 5431  (class class class)co 6103   CCcc 9292   0cc0 9294   1c1 9295   + caddc 9297   < clt 9430   NNcn 10334   ZZcz 10658   ZZ>=cuz 10873   ...cfz 11449   seqcseq 11818   #chash 12115

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-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372

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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-1o 6932  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-fin 7326  df-sup 7703  df-card 8121  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-n0 10592  df-z 10659  df-uz 10874  df-fz 11450  df-seq 11819  df-hash 12116

This theorem is referenced by:  isercoll  13157