Theorem sumrblem 13854

Description: Lemma for sumrb 13856. (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 9834	. . 3 |- ( n e. CC -> ( 0 + n ) = n )
2	1	adantl 473	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. CC ) -> ( 0 + n ) = n )
3		0cnd 9654	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> 0 e. CC )
4		sumrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5	4	adantr 472	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		iftrue 3878	. . . . . . . . . 10 |- ( k e. A -> if ( k e. A , B , 0 ) = B )
7	6	adantl 473	. . . . . . . . 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 2549	. . . . . . . 8 |- ( ( ph /\ k e. A ) -> if ( k e. A , B , 0 ) e. CC )
10	9	ex 441	. . . . . . 7 |- ( ph -> ( k e. A -> if ( k e. A , B , 0 ) e. CC ) )
11		iffalse 3881	. . . . . . . 8 |- ( -. k e. A -> if ( k e. A , B , 0 ) = 0 )
12		0cn 9653	. . . . . . . 8 |- 0 e. CC
13	11, 12	syl6eqel 2557	. . . . . . 7 |- ( -. k e. A -> if ( k e. A , B , 0 ) e. CC )
14	10, 13	pm2.61d1 164	. . . . . 6 |- ( ph -> if ( k e. A , B , 0 ) e. CC )
15	14	adantr 472	. . . . 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 6061	. . . 4 |- ( ph -> F : ZZ --> CC )
18	17	adantr 472	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> F : ZZ --> CC )
19		eluzelz 11192	. . . . 5 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
20	4, 19	syl 17	. . . 4 |- ( ph -> N e. ZZ )
21	20	adantr 472	. . 3 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ZZ )
22	18, 21	ffvelrnd 6038	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( F ` N ) e. CC )
23		elfzelz 11826	. . . . 5 |- ( n e. ( M ... ( N - 1 ) ) -> n e. ZZ )
24	23	adantl 473	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ZZ )
25		simplr 770	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` N ) )
26	20	zcnd 11064	. . . . . . . . 9 |- ( ph -> N e. CC )
27	26	ad2antrr 740	. . . . . . . 8 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> N e. CC )
28		ax-1cn 9615	. . . . . . . 8 |- 1 e. CC
29		npcan 9904	. . . . . . . 8 |- ( ( N e. CC /\ 1 e. CC ) -> ( ( N - 1 ) + 1 ) = N )
30	27, 28, 29	sylancl 675	. . . . . . 7 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
31	30	fveq2d 5883	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
32	25, 31	sseqtr4d 3455	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
33		fznuz 11902	. . . . . 6 |- ( n e. ( M ... ( N - 1 ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
34	33	adantl 473	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
35	32, 34	ssneldd 3421	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. A )
36	24, 35	eldifd 3401	. . 3 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( ZZ \ A ) )
37		fveq2 5879	. . . . 5 |- ( k = n -> ( F ` k ) = ( F ` n ) )
38	37	eqeq1d 2473	. . . 4 |- ( k = n -> ( ( F ` k ) = 0 <-> ( F ` n ) = 0 ) )
39		eldifi 3544	. . . . . 6 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
40		eldifn 3545	. . . . . . . 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 2557	. . . . . 6 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 0 ) e. CC )
43	16	fvmpt2 5972	. . . . . 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 673	. . . . 5 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 0 ) )
45	44, 41	eqtrd 2505	. . . 4 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 0 )
46	38, 45	vtoclga 3099	. . 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 12296	1 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( + , F ) |` ( ZZ>= ` N ) ) = seq N ( + , F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   \ cdif 3387   C_ wss 3390   ifcif 3872   |-> cmpt 4454   |` cres 4841   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   0cc0 9557   1c1 9558   + caddc 9560   - cmin 9880   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810   seqcseq 12251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-seq 12252

This theorem is referenced by:  sumrb  13856