Theorem isercolllem3 13140

Description: Lemma for isercoll 13141. (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 9548	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 463	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		addid1 9545	. . 3 |- ( n e. CC -> ( n + 0 ) = n )
4	3	adantl 463	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. CC ) -> ( n + 0 ) = n )
5		addcl 9360	. . 3 |- ( ( n e. CC /\ k e. CC ) -> ( n + k ) e. CC )
6	5	adantl 463	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ ( n e. CC /\ k e. CC ) ) -> ( n + k ) e. CC )
7		0cnd 9375	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 0 e. CC )
8		cnvimass 5186	. . . . 5 |- ( `' G " ( M ... N ) ) C_ dom G
9		isercoll.g	. . . . . . 7 |- ( ph -> G : NN --> Z )
10	9	adantr 462	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> G : NN --> Z )
11		fdm 5560	. . . . . 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 3401	. . . 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 13138	. . . 4 |- ( ( ph /\ ( `' G " ( M ... N ) ) C_ NN ) -> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( `' G " ( M ... N ) ) , ( G " ( `' G " ( M ... N ) ) ) ) )
18	13, 17	syldan 467	. . 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 13139	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) = ( `' G " ( M ... N ) ) )
20		isoeq4 6010	. . . 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 3552	. . . . 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 10329	. . . . . . 7 |- 1 e. NN
27	26	a1i 11	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. NN )
28		ffvelrn 5838	. . . . . . . . . 10 |- ( ( G : NN --> Z /\ 1 e. NN ) -> ( G ` 1 ) e. Z )
29	9, 26, 28	sylancl 657	. . . . . . . . 9 |- ( ph -> ( G ` 1 ) e. Z )
30	29, 14	syl6eleq 2531	. . . . . . . 8 |- ( ph -> ( G ` 1 ) e. ( ZZ>= ` M ) )
31	30	adantr 462	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G ` 1 ) e. ( ZZ>= ` M ) )
32		simpr 458	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> N e. ( ZZ>= ` ( G ` 1 ) ) )
33		elfzuzb 11443	. . . . . . 7 |- ( ( G ` 1 ) e. ( M ... N ) <-> ( ( G ` 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) )
34	31, 32, 33	sylanbrc 659	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G ` 1 ) e. ( M ... N ) )
35		ffn 5556	. . . . . . 7 |- ( G : NN --> Z -> G Fn NN )
36		elpreima 5820	. . . . . . 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 908	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. ( `' G " ( M ... N ) ) )
39		ne0i 3640	. . . . 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 2624	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
42		imadisj 5185	. . . 4 |- ( ( G " ( `' G " ( M ... N ) ) ) = (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = (/) )
43	42	necon3bii 2638	. . 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 5558	. . . 4 |- ( G : NN --> Z -> Fun G )
46		funimacnv 5487	. . . 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 3567	. . . 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 3387	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) C_ ( M ... N ) )
51		simpl 454	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ph )
52		elfzuz 11445	. . . 4 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
53	52, 14	syl6eleqr 2532	. . 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 474	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( M ... N ) ) -> ( F ` n ) e. CC )
56	47	difeq2d 3471	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) )
57		difin 3584	. . . . . 6 |- ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) = ( ( M ... N ) \ ran G )
58	56, 57	syl6eq 2489	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ran G ) )
59	53	ssriv 3357	. . . . . 6 |- ( M ... N ) C_ Z
60		ssdif 3488	. . . . . 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 3387	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) C_ ( Z \ ran G ) )
63	62	sselda 3353	. . 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 709	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( Z \ ran G ) ) -> ( F ` n ) = 0 )
66	63, 65	syldan 467	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) ) -> ( F ` n ) = 0 )
67		elfznn 11474	. . . 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 474	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( H ` k ) = ( F ` ( G ` k ) ) )
70	19	eleq2d 2508	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) <-> k e. ( `' G " ( M ... N ) ) ) )
71	70	biimpa 481	. . . . 5 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> k e. ( `' G " ( M ... N ) ) )
72		fvres 5701	. . . . 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 5692	. . 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 2476	. 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 12213	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 1364   e. wcel 1761   =/= wne 2604   \ cdif 3322   i^i cin 3324   C_ wss 3325   (/)c0 3634   class class class wbr 4289   `'ccnv 4835   dom cdm 4836   ran crn 4837   |` cres 4838   "cima 4839   Fun wfun 5409   Fn wfn 5410   -->wf 5411   ` cfv 5415   Isom wiso 5416  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279   + caddc 9281   < clt 9414   NNcn 10318   ZZcz 10642   ZZ>=cuz 10857   ...cfz 11433   seqcseq 11802   #chash 12099

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 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356

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 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-1o 6916  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-sup 7687  df-card 8105  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-hash 12100

This theorem is referenced by:  isercoll  13141