Theorem sumrblem 13513

Description: Lemma for sumrb 13515. (Contributed by Mario Carneiro, 12-Aug-2013.)

Hypotheses

Ref	Expression
summo.1	|- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
summo.2	|- ( ( ph /\ k e. A ) -> B e. CC )
sumrb.3	|- ( ph -> N e. ( ZZ>= ` M ) )

Assertion

Ref	Expression
sumrblem	|- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )

Distinct variable groups:   A, k   k, F   k, N   ph, k   k, M

Allowed substitution hint:   B( k)

Proof of Theorem sumrblem

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		addid2 9774	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 466	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		0cnd 9601	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> 0 e. CC )
4		sumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5	4	adantr 465	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		iftrue 3951	. . . . . . . . . 10 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
7	6	adantl 466	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) = B )
8		summo.2	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> B e. CC )
9	7, 8	eqeltrd 2555	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
10	9	ex 434	. . . . . . 7 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
11		iffalse 3954	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
12		0cn 9600	. . . . . . . 8 |- 0 e. CC
13	11, 12	syl6eqel 2563	. . . . . . 7 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
14	10, 13	pm2.61d1 159	. . . . . 6 |- ( ph -> if ( k e. A , B , 0 ) e. CC )
15	14	adantr 465	. . . . 5 |- ( ( ph /\ k e. ZZ ) -> if ( k e. A , B , 0 ) e. CC )
16		summo.1	. . . . 5 |- F = ( k e. ZZ |-> if ( k e. A , B , 0 ) )
17	15, 16	fmptd 6056	. . . 4 |- ( ph -> F : ZZ --> CC )
18	17	adantr 465	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> F : ZZ --> CC )
19		eluzelz 11103	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
20	4, 19	syl 16	. . . 4 |- ( ph -> N e. ZZ )
21	20	adantr 465	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ZZ )
22	18, 21	ffvelrnd 6033	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( F ` N ) e. CC )
23		elfzelz 11700	. . . . 5 |- ( n e. ( M ... ( N - 1 ) ) -> n e. ZZ )
24	23	adantl 466	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ZZ )
25		simplr 754	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` N ) )
26	20	zcnd 10979	. . . . . . . . 9 |- ( ph -> N e. CC )
27	26	ad2antrr 725	. . . . . . . 8 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> N e. CC )
28		ax-1cn 9562	. . . . . . . 8 |- 1 e. CC
29		npcan 9841	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
30	27, 28, 29	sylancl 662	. . . . . . 7 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
31	30	fveq2d 5876	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
32	25, 31	sseqtr4d 3546	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
33		fznuz 11772	. . . . . 6 |- ( n e. ( M ... ( N - 1 ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
34	33	adantl 466	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
35	32, 34	ssneldd 3512	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. A )
36	24, 35	eldifd 3492	. . 3 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( ZZ \ A ) )
37		fveq2 5872	. . . . 5 |- ( k = n -> ( F ` k ) = ( F ` n ) )
38	37	eqeq1d 2469	. . . 4 |- ( k = n -> ( ( F ` k ) = 0 <-> ( F ` n ) = 0 ) )
39		eldifi 3631	. . . . . 6 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
40		eldifn 3632	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> -. k e. A )
41	40, 11	syl 16	. . . . . . 7 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
42	41, 12	syl6eqel 2563	. . . . . 6 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
43	16	fvmpt2 5964	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 0 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
44	39, 42, 43	syl2anc 661	. . . . 5 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
45	44, 41	eqtrd 2508	. . . 4 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
46	38, 45	vtoclga 3182	. . 3 |- ( n e. ( ZZ \ A ) -> ( F ` n ) = 0 )
47	36, 46	syl 16	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) = 0 )
48	2, 3, 5, 22, 47	seqid 12132	1 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1379   e. wcel 1767   \ cdif 3478   C_ wss 3481   ifcif 3945   |-> cmpt 4511   |` cres 5007   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   1c1 9505   + caddc 9507   - cmin 9817   ZZcz 10876   ZZ>=cuz 11094   ...cfz 11684   seqcseq 12087

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-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581

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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-seq 12088

This theorem is referenced by:  sumrb  13515