Theorem sumrblem 13789

Description: Lemma for sumrb 13791. (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 9821	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 468	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		0cnd 9641	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> 0 e. CC )
4		sumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5	4	adantr 467	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		iftrue 3889	. . . . . . . . . 10 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
7	6	adantl 468	. . . . . . . . 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 2531	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
10	9	ex 436	. . . . . . 7 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
11		iffalse 3892	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
12		0cn 9640	. . . . . . . 8 |- 0 e. CC
13	11, 12	syl6eqel 2539	. . . . . . 7 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
14	10, 13	pm2.61d1 163	. . . . . 6 |- ( ph -> if ( k e. A , B , 0 ) e. CC )
15	14	adantr 467	. . . . 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 6051	. . . 4 |- ( ph -> F : ZZ --> CC )
18	17	adantr 467	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> F : ZZ --> CC )
19		eluzelz 11175	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
20	4, 19	syl 17	. . . 4 |- ( ph -> N e. ZZ )
21	20	adantr 467	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ZZ )
22	18, 21	ffvelrnd 6028	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( F ` N ) e. CC )
23		elfzelz 11807	. . . . 5 |- ( n e. ( M ... ( N - 1 ) ) -> n e. ZZ )
24	23	adantl 468	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ZZ )
25		simplr 763	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` N ) )
26	20	zcnd 11048	. . . . . . . . 9 |- ( ph -> N e. CC )
27	26	ad2antrr 733	. . . . . . . 8 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> N e. CC )
28		ax-1cn 9602	. . . . . . . 8 |- 1 e. CC
29		npcan 9889	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
30	27, 28, 29	sylancl 669	. . . . . . 7 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
31	30	fveq2d 5874	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
32	25, 31	sseqtr4d 3471	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
33		fznuz 11883	. . . . . 6 |- ( n e. ( M ... ( N - 1 ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
34	33	adantl 468	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
35	32, 34	ssneldd 3437	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. A )
36	24, 35	eldifd 3417	. . 3 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( ZZ \ A ) )
37		fveq2 5870	. . . . 5 |- ( k = n -> ( F ` k ) = ( F ` n ) )
38	37	eqeq1d 2455	. . . 4 |- ( k = n -> ( ( F ` k ) = 0 <-> ( F ` n ) = 0 ) )
39		eldifi 3557	. . . . . 6 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
40		eldifn 3558	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> -. k e. A )
41	40, 11	syl 17	. . . . . . 7 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
42	41, 12	syl6eqel 2539	. . . . . 6 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
43	16	fvmpt2 5962	. . . . . 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 667	. . . . 5 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
45	44, 41	eqtrd 2487	. . . 4 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
46	38, 45	vtoclga 3115	. . 3 |- ( n e. ( ZZ \ A ) -> ( F ` n ) = 0 )
47	36, 46	syl 17	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) = 0 )
48	2, 3, 5, 22, 47	seqid 12265	1 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1446   e. wcel 1889   \ cdif 3403   C_ wss 3406   ifcif 3883   |-> cmpt 4464   |` cres 4839   -->wf 5581   ` cfv 5585  (class class class)co 6295   CCcc 9542   0cc0 9544   1c1 9545   + caddc 9547   - cmin 9865   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791   seqcseq 12220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-seq 12221

This theorem is referenced by:  sumrb  13791