Theorem prodrblem 14659

Description: Lemma for prodrb 14662. (Contributed by Scott Fenton, 4-Dec-2017.)

Hypotheses

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

Assertion

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

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

Allowed substitution hints:   B( k)   M( k)   N( k)

Proof of Theorem prodrblem

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulid2 10038	. . 3 |- ( n e. CC -> ( 1 x. n ) = n )
2	1	adantl 482	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. CC ) -> ( 1 x. n ) = n )
3		1cnd 10056	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> 1 e. CC )
4		prodrb.3	. . 3 |- ( ph -> N e. ( ZZ>= ` M ) )
5	4	adantr 481	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. ( ZZ>= ` M ) )
6		iftrue 4092	. . . . . . . . 9 |- ( k e. A -> if ( k e. A , B , 1 ) = B )
7	6	adantl 482	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ k e. A ) -> if ( k e. A , B , 1 ) = B )
8		prodmo.2	. . . . . . . . 9 |- ( ( ph /\ k e. A ) -> B e. CC )
9	8	adantlr 751	. . . . . . . 8 |- ( ( ( ph /\ k e. ZZ ) /\ k e. A ) -> B e. CC )
10	7, 9	eqeltrd 2701	. . . . . . 7 |- ( ( ( ph /\ k e. ZZ ) /\ k e. A ) -> if ( k e. A , B , 1 ) e. CC )
11	10	ex 450	. . . . . 6 |- ( ( ph /\ k e. ZZ ) -> ( k e. A -> if ( k e. A , B , 1 ) e. CC ) )
12		iffalse 4095	. . . . . . 7 |- ( -. k e. A -> if ( k e. A , B , 1 ) = 1 )
13		ax-1cn 9994	. . . . . . 7 |- 1 e. CC
14	12, 13	syl6eqel 2709	. . . . . 6 |- ( -. k e. A -> if ( k e. A , B , 1 ) e. CC )
15	11, 14	pm2.61d1 171	. . . . 5 |- ( ( ph /\ k e. ZZ ) -> if ( k e. A , B , 1 ) e. CC )
16		prodmo.1	. . . . 5 |- F = ( k e. ZZ |-> if ( k e. A , B , 1 ) )
17	15, 16	fmptd 6385	. . . 4 |- ( ph -> F : ZZ --> CC )
18		uzssz 11707	. . . . 5 |- ( ZZ>= ` M ) C_ ZZ
19	18, 4	sseldi 3601	. . . 4 |- ( ph -> N e. ZZ )
20	17, 19	ffvelrnd 6360	. . 3 |- ( ph -> ( F ` N ) e. CC )
21	20	adantr 481	. 2 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( F ` N ) e. CC )
22		elfzelz 12342	. . . . 5 |- ( n e. ( M ... ( N - 1 ) ) -> n e. ZZ )
23	22	adantl 482	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ZZ )
24		simplr 792	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` N ) )
25	19	zcnd 11483	. . . . . . . . . 10 |- ( ph -> N e. CC )
26	25	adantr 481	. . . . . . . . 9 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> N e. CC )
27	26	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> N e. CC )
28		1cnd 10056	. . . . . . . 8 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> 1 e. CC )
29	27, 28	npcand 10396	. . . . . . 7 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ( N - 1 ) + 1 ) = N )
30	29	fveq2d 6195	. . . . . 6 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( ZZ>= ` ( ( N - 1 ) + 1 ) ) = ( ZZ>= ` N ) )
31	24, 30	sseqtr4d 3642	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> A C_ ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
32		fznuz 12422	. . . . . 6 |- ( n e. ( M ... ( N - 1 ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
33	32	adantl 482	. . . . 5 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. ( ZZ>= ` ( ( N - 1 ) + 1 ) ) )
34	31, 33	ssneldd 3606	. . . 4 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> -. n e. A )
35	23, 34	eldifd 3585	. . 3 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> n e. ( ZZ \ A ) )
36		fveq2 6191	. . . . 5 |- ( k = n -> ( F ` k ) = ( F ` n ) )
37	36	eqeq1d 2624	. . . 4 |- ( k = n -> ( ( F ` k ) = 1 <-> ( F ` n ) = 1 ) )
38		eldifi 3732	. . . . . 6 |- ( k e. ( ZZ \ A ) -> k e. ZZ )
39		eldifn 3733	. . . . . . . 8 |- ( k e. ( ZZ \ A ) -> -. k e. A )
40	39, 12	syl 17	. . . . . . 7 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 1 ) = 1 )
41	40, 13	syl6eqel 2709	. . . . . 6 |- ( k e. ( ZZ \ A ) -> if ( k e. A , B , 1 ) e. CC )
42	16	fvmpt2 6291	. . . . . 6 |- ( ( k e. ZZ /\ if ( k e. A , B , 1 ) e. CC ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
43	38, 41, 42	syl2anc 693	. . . . 5 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = if ( k e. A , B , 1 ) )
44	43, 40	eqtrd 2656	. . . 4 |- ( k e. ( ZZ \ A ) -> ( F ` k ) = 1 )
45	37, 44	vtoclga 3272	. . 3 |- ( n e. ( ZZ \ A ) -> ( F ` n ) = 1 )
46	35, 45	syl 17	. 2 |- ( ( ( ph /\ A C_ ( ZZ>= ` N ) ) /\ n e. ( M ... ( N - 1 ) ) ) -> ( F ` n ) = 1 )
47	2, 3, 5, 21, 46	seqid 12846	1 |- ( ( ph /\ A C_ ( ZZ>= ` N ) ) -> ( seq M ( x. , F ) |` ( ZZ>= ` N ) ) = seq N ( x. , F ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   ifcif 4086   |-> cmpt 4729   |` cres 5116   ` cfv 5888  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  prodrblem2  14661