Theorem sumrblem 13193

Description: Lemma for sumrb 13195. (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 9557	. . 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 9384	. 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 3802	. . . . . . . . . 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 2517	. . . . . . . 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 3804	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
12		0cn 9383	. . . . . . . 8 |- 0 e. CC
13	11, 12	syl6eqel 2531	. . . . . . 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 5872	. . . 4 |- ( ph -> F : ZZ --> CC )
18	17	adantr 465	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> F : ZZ --> CC )
19		eluzelz 10875	. . . . 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 5849	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( F ` N ) e. CC )
23		elfzelz 11458	. . . . 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 10753	. . . . . . . . 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 9345	. . . . . . . 8 |- 1 e. CC
29		npcan 9624	. . . . . . . 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 5700	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
32	25, 31	sseqtr4d 3398	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
33		fznuz 11547	. . . . . 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 3364	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. A )
36	24, 35	eldifd 3344	. . 3 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( ZZ \ A ) )
37		fveq2 5696	. . . . 5 |- ( k = n -> ( F ` k ) = ( F ` n ) )
38	37	eqeq1d 2451	. . . 4 |- ( k = n -> ( ( F ` k ) = 0 <-> ( F ` n ) = 0 ) )
39		eldifi 3483	. . . . . 6 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
40		eldifn 3484	. . . . . . . 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 2531	. . . . . 6 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
43	16	fvmpt2 5786	. . . . . 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 2475	. . . 4 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
46	38, 45	vtoclga 3041	. . 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 11856	1 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1369   e. wcel 1756   \ cdif 3330   C_ wss 3333   ifcif 3796   e. cmpt 4355   |` cres 4847   -->wf 5419   ` cfv 5423  (class class class)co 6096   CCcc 9285   0cc0 9287   1c1 9288   + caddc 9290   - cmin 9600   ZZcz 10651   ZZ>=cuz 10866   ...cfz 11442   seqcseq 11811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812

This theorem is referenced by:  sumrb  13195