Theorem isercolllem3 13445

Description: Lemma for isercoll 13446. (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 9758	. . 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 9755	. . 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 9570	. . 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 9585	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 0 e. CC )
8		cnvimass 5355	. . . . 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 5733	. . . . . 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 3552	. . . 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 13443	. . . 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 13444	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) = ( `' G " ( M ... N ) ) )
20		isoeq4 6204	. . . 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 3703	. . . . 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 10543	. . . . . . 7 |- 1 e. NN
27	26	a1i 11	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. NN )
28		ffvelrn 6017	. . . . . . . . . 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 2565	. . . . . . . 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 11678	. . . . . . 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 5729	. . . . . . 7 |- ( G : NN --> Z -> G Fn NN )
36		elpreima 5999	. . . . . . 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 920	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. ( `' G " ( M ... N ) ) )
39		ne0i 3791	. . . . 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 2760	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
42		imadisj 5354	. . . 4 |- ( ( G " ( `' G " ( M ... N ) ) ) = (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = (/) )
43	42	necon3bii 2735	. . 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 5731	. . . 4 |- ( G : NN --> Z -> Fun G )
46		funimacnv 5658	. . . 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 3718	. . . 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 3538	. 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 11680	. . . 4 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
53	52, 14	syl6eleqr 2566	. . 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 3622	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) )
57		difin 3735	. . . . . 6 |- ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) = ( ( M ... N ) \ ran G )
58	56, 57	syl6eq 2524	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ran G ) )
59	53	ssriv 3508	. . . . . 6 |- ( M ... N ) C_ Z
60		ssdif 3639	. . . . . 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 3538	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) C_ ( Z \ ran G ) )
63	62	sselda 3504	. . 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 11710	. . . 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 2537	. . . . . 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 5878	. . . . 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 5868	. . 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 2511	. 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 12473	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 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   i^i cin 3475   C_ wss 3476   (/)c0 3785   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   ran crn 5000   |` cres 5001   "cima 5002   Fun wfun 5580   Fn wfn 5581   -->wf 5582   ` cfv 5586   Isom wiso 5587  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489   + caddc 9491   < clt 9624   NNcn 10532   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668   seqcseq 12070   #chash 12367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-sup 7897  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-seq 12071  df-hash 12368

This theorem is referenced by:  isercoll  13446