Theorem isercolllem3 14397

Description: Lemma for isercoll 14398. (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 10219	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 482	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		addid1 10216	. . 3 |- ( n e. CC -> ( n + 0 ) = n )
4	3	adantl 482	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. CC ) -> ( n + 0 ) = n )
5		addcl 10018	. . 3 |- ( ( n e. CC /\ k e. CC ) -> ( n + k ) e. CC )
6	5	adantl 482	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ ( n e. CC /\ k e. CC ) ) -> ( n + k ) e. CC )
7		0cnd 10033	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 0 e. CC )
8		cnvimass 5485	. . . . 5 |- ( `' G " ( M ... N ) ) C_ dom G
9		isercoll.g	. . . . . . 7 |- ( ph -> G : NN --> Z )
10	9	adantr 481	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> G : NN --> Z )
11		fdm 6051	. . . . . 6 |- ( G : NN --> Z -> dom G = NN )
12	10, 11	syl 17	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> dom G = NN )
13	8, 12	syl5sseq 3653	. . . 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 14395	. . . 4 |- ( ( ph /\ ( `' G " ( M ... N ) ) C_ NN ) -> ( G |` ( `' G " ( M ... N ) ) ) Isom < , < ( ( `' G " ( M ... N ) ) , ( G " ( `' G " ( M ... N ) ) ) ) )
18	13, 17	syldan 487	. . 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 14396	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) = ( `' G " ( M ... N ) ) )
20		isoeq4 6570	. . . 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 17	. . 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 247	. 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		sseqin2 3817	. . . . 5 |- ( ( `' G " ( M ... N ) ) C_ dom G <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = ( `' G " ( M ... N ) ) )
25	23, 24	sylib 208	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) = ( `' G " ( M ... N ) ) )
26		1nn 11031	. . . . . . 7 |- 1 e. NN
27	26	a1i 11	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. NN )
28		ffvelrn 6357	. . . . . . . . . 10 |- ( ( G : NN --> Z /\ 1 e. NN ) -> ( G ` 1 ) e. Z )
29	9, 26, 28	sylancl 694	. . . . . . . . 9 |- ( ph -> ( G ` 1 ) e. Z )
30	29, 14	syl6eleq 2711	. . . . . . . 8 |- ( ph -> ( G ` 1 ) e. ( ZZ>= ` M ) )
31	30	adantr 481	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G ` 1 ) e. ( ZZ>= ` M ) )
32		simpr 477	. . . . . . 7 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> N e. ( ZZ>= ` ( G ` 1 ) ) )
33		elfzuzb 12336	. . . . . . 7 |- ( ( G ` 1 ) e. ( M ... N ) <-> ( ( G ` 1 ) e. ( ZZ>= ` M ) /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) )
34	31, 32, 33	sylanbrc 698	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G ` 1 ) e. ( M ... N ) )
35		ffn 6045	. . . . . . 7 |- ( G : NN --> Z -> G Fn NN )
36		elpreima 6337	. . . . . . 7 |- ( G Fn NN -> ( 1 e. ( `' G " ( M ... N ) ) <-> ( 1 e. NN /\ ( G ` 1 ) e. ( M ... N ) ) ) )
37	10, 35, 36	3syl 18	. . . . . 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 957	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> 1 e. ( `' G " ( M ... N ) ) )
39		ne0i 3921	. . . . 5 |- ( 1 e. ( `' G " ( M ... N ) ) -> ( `' G " ( M ... N ) ) =/= (/) )
40	38, 39	syl 17	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( `' G " ( M ... N ) ) =/= (/) )
41	25, 40	eqnetrd 2861	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
42		imadisj 5484	. . . 4 |- ( ( G " ( `' G " ( M ... N ) ) ) = (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) = (/) )
43	42	necon3bii 2846	. . 3 |- ( ( G " ( `' G " ( M ... N ) ) ) =/= (/) <-> ( dom G i^i ( `' G " ( M ... N ) ) ) =/= (/) )
44	41, 43	sylibr 224	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) =/= (/) )
45		ffun 6048	. . . 4 |- ( G : NN --> Z -> Fun G )
46		funimacnv 5970	. . . 4 |- ( Fun G -> ( G " ( `' G " ( M ... N ) ) ) = ( ( M ... N ) i^i ran G ) )
47	10, 45, 46	3syl 18	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) = ( ( M ... N ) i^i ran G ) )
48		inss1 3833	. . . 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 3639	. 2 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( G " ( `' G " ( M ... N ) ) ) C_ ( M ... N ) )
51		simpl 473	. . 3 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ph )
52		elfzuz 12338	. . . 4 |- ( n e. ( M ... N ) -> n e. ( ZZ>= ` M ) )
53	52, 14	syl6eleqr 2712	. . 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 494	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( M ... N ) ) -> ( F ` n ) e. CC )
56	47	difeq2d 3728	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) )
57		difin 3861	. . . . . 6 |- ( ( M ... N ) \ ( ( M ... N ) i^i ran G ) ) = ( ( M ... N ) \ ran G )
58	56, 57	syl6eq 2672	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) = ( ( M ... N ) \ ran G ) )
59	53	ssriv 3607	. . . . . 6 |- ( M ... N ) C_ Z
60		ssdif 3745	. . . . . 6 |- ( ( M ... N ) C_ Z -> ( ( M ... N ) \ ran G ) C_ ( Z \ ran G ) )
61	59, 60	mp1i 13	. . . . 5 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ran G ) C_ ( Z \ ran G ) )
62	58, 61	eqsstrd 3639	. . . 4 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) C_ ( Z \ ran G ) )
63	62	sselda 3603	. . 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 751	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( Z \ ran G ) ) -> ( F ` n ) = 0 )
66	63, 65	syldan 487	. 2 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ n e. ( ( M ... N ) \ ( G " ( `' G " ( M ... N ) ) ) ) ) -> ( F ` n ) = 0 )
67		elfznn 12370	. . . 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 494	. . 3 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( H ` k ) = ( F ` ( G ` k ) ) )
70	19	eleq2d 2687	. . . . . 6 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) <-> k e. ( `' G " ( M ... N ) ) ) )
71	70	biimpa 501	. . . . 5 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> k e. ( `' G " ( M ... N ) ) )
72		fvres 6207	. . . . 5 |- ( k e. ( `' G " ( M ... N ) ) -> ( ( G |` ( `' G " ( M ... N ) ) ) ` k ) = ( G ` k ) )
73	71, 72	syl 17	. . . 4 |- ( ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) /\ k e. ( 1 ... ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) ) -> ( ( G |` ( `' G " ( M ... N ) ) ) ` k ) = ( G ` k ) )
74	73	fveq2d 6195	. . 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 2659	. 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 13249	1 |- ( ( ph /\ N e. ( ZZ>= ` ( G ` 1 ) ) ) -> ( seq M ( + , F ) ` N ) = ( seq 1 ( + , H ) ` ( # ` ( G " ( `' G " ( M ... N ) ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   class class class wbr 4653   `'ccnv 5113   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-hash 13118

This theorem is referenced by:  isercoll  14398