Theorem sumrblem 12460

Description: Lemma for sumrb 12462. (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 9205	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 453	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		0cn 9040	. . 3 |- 0 e. CC
4	3	a1i 11	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> 0 e. CC )
5		sumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
6	5	adantr 452	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
7		iftrue 3705	. . . . . . . . . 10 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
8	7	adantl 453	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) = B )
9		summo.2	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> B e. CC )
10	8, 9	eqeltrd 2478	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
11	10	ex 424	. . . . . . 7 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
12		iffalse 3706	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
13	12, 3	syl6eqel 2492	. . . . . . 7 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
14	11, 13	pm2.61d1 153	. . . . . 6 |- ( ph -> if ( k e. A , B , 0 ) e. CC )
15	14	adantr 452	. . . . 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 5852	. . . 4 |- ( ph -> F : ZZ --> CC )
18	17	adantr 452	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> F : ZZ --> CC )
19		eluzelz 10452	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
20	5, 19	syl 16	. . . 4 |- ( ph -> N e. ZZ )
21	20	adantr 452	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ZZ )
22	18, 21	ffvelrnd 5830	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( F ` N ) e. CC )
23		elfzelz 11015	. . . . 5 |- ( n e. ( M ... ( N - 1 ) ) -> n e. ZZ )
24	23	adantl 453	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ZZ )
25		simplr 732	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` N ) )
26	20	zcnd 10332	. . . . . . . . 9 |- ( ph -> N e. CC )
27	26	ad2antrr 707	. . . . . . . 8 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> N e. CC )
28		ax-1cn 9004	. . . . . . . 8 |- 1 e. CC
29		npcan 9270	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
30	27, 28, 29	sylancl 644	. . . . . . 7 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
31	30	fveq2d 5691	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
32	25, 31	sseqtr4d 3345	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
33		fznuz 11084	. . . . . 6 |- ( n e. ( M ... ( N - 1 ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
34	33	adantl 453	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
35	32, 34	ssneldd 3311	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. A )
36	24, 35	eldifd 3291	. . 3 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( ZZ \ A ) )
37		fveq2 5687	. . . . 5 |- ( k = n -> ( F ` k ) = ( F ` n ) )
38	37	eqeq1d 2412	. . . 4 |- ( k = n -> ( ( F ` k ) = 0 <-> ( F ` n ) = 0 ) )
39		eldifi 3429	. . . . . 6 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
40		eldifn 3430	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> -. k e. A )
41	40, 12	syl 16	. . . . . . 7 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) = 0 )
42	41, 3	syl6eqel 2492	. . . . . 6 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
43	16	fvmpt2 5771	. . . . . 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 643	. . . . 5 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
45	44, 41	eqtrd 2436	. . . 4 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
46	38, 45	vtoclga 2977	. . 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, 4, 6, 22, 47	seqid 11323	1 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   \ cdif 3277   C_ wss 3280   ifcif 3699   e. cmpt 4226   |` cres 4839   -->wf 5409   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   + caddc 8949   - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   seq cseq 11278

This theorem is referenced by:  sumrb  12462

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  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-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-seq 11279