Theorem fprodm1 11605

Description: Separate out the last term in a finite product. (Contributed by Scott Fenton, 16-Dec-2017.)

Hypotheses

Ref	Expression
fprodm1.1	|- ( ph -> N e. ( ZZ>= ` M ) )
fprodm1.2	|- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
fprodm1.3	|- ( k = N -> A = B )

Assertion

Ref	Expression
fprodm1	|- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. B ) )

Distinct variable groups:   B, k   ph, k   k, M   k, N

Allowed substitution hint:   A( k)

Proof of Theorem fprodm1

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzp1nel 10103	. . . . 5 |- -. ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) )
2		fprodm1.1	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9536	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
5	4	zcnd 9375	. . . . . . 7 |- ( ph -> N e. CC )
6		1cnd 7972	. . . . . . 7 |- ( ph -> 1 e. CC )
7	5, 6	npcand 8271	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
8	7	eleq1d 2246	. . . . 5 |- ( ph -> ( ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) ) <-> N e. ( M ... ( N - 1 ) ) ) )
9	1, 8	mtbii 674	. . . 4 |- ( ph -> -. N e. ( M ... ( N - 1 ) ) )
10		disjsn 3654	. . . 4 |- ( ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) <-> -. N e. ( M ... ( N - 1 ) ) )
11	9, 10	sylibr 134	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9532	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	2, 12	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
14		peano2zm 9290	. . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . 6 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9375	. . . . . . . . 9 |- ( ph -> M e. CC )
17	16, 6	npcand 8271	. . . . . . . 8 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
18	17	fveq2d 5519	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
19	2, 18	eleqtrrd 2257	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
20		eluzp1m1 9550	. . . . . 6 |- ( ( ( M - 1 ) e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
21	15, 19, 20	syl2anc 411	. . . . 5 |- ( ph -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
22		fzsuc2 10078	. . . . 5 |- ( ( M e. ZZ /\ ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) ) -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
23	13, 21, 22	syl2anc 411	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
24	7	oveq2d 5890	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
25	7	sneqd 3605	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
26	25	uneq2d 3289	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
27	23, 24, 26	3eqtr3d 2218	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
28	13, 4	fzfigd 10430	. . 3 |- ( ph -> ( M ... N ) e. Fin )
29		elfzelz 10024	. . . . . 6 |- ( j e. ( M ... N ) -> j e. ZZ )
30	29	adantl 277	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> j e. ZZ )
31	13	adantr 276	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> M e. ZZ )
32	4	adantr 276	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> N e. ZZ )
33		peano2zm 9290	. . . . . 6 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
34	32, 33	syl 14	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> ( N - 1 ) e. ZZ )
35		fzdcel 10039	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ ( N - 1 ) e. ZZ ) -> DECID j e. ( M ... ( N - 1 ) ) )
36	30, 31, 34, 35	syl3anc 1238	. . . 4 |- ( ( ph /\ j e. ( M ... N ) ) -> DECID j e. ( M ... ( N - 1 ) ) )
37	36	ralrimiva 2550	. . 3 |- ( ph -> A. j e. ( M ... N )DECID j e. ( M ... ( N - 1 ) ) )
38		fprodm1.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
39	11, 27, 28, 37, 38	fprodsplitdc 11603	. 2 |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. prod_ k e. { N } A ) )
40		fprodm1.3	. . . . . 6 |- ( k = N -> A = B )
41	40	eleq1d 2246	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
42	38	ralrimiva 2550	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
43		eluzfz2 10031	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> N e. ( M ... N ) )
44	2, 43	syl 14	. . . . 5 |- ( ph -> N e. ( M ... N ) )
45	41, 42, 44	rspcdva 2846	. . . 4 |- ( ph -> B e. CC )
46	40	prodsn 11600	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ B e. CC ) -> prod_ k e. { N } A = B )
47	2, 45, 46	syl2anc 411	. . 3 |- ( ph -> prod_ k e. { N } A = B )
48	47	oveq2d 5890	. 2 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. prod_ k e. { N } A ) = ( prod_ k e. ( M ... ( N - 1 ) ) A x. B ) )
49	39, 48	eqtrd 2210	1 |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 834   = wceq 1353   e. wcel 2148   u. cun 3127   i^i cin 3128   (/)c0 3422   {csn 3592   ` cfv 5216  (class class class)co 5874   CCcc 7808   1c1 7811   + caddc 7813   x. cmul 7815   - cmin 8127   ZZcz 9252   ZZ>=cuz 9527   ...cfz 10007   prod_cprod 11557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-3 8978  df-4 8979  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-fz 10008  df-fzo 10142  df-seqfrec 10445  df-exp 10519  df-ihash 10755  df-cj 10850  df-re 10851  df-im 10852  df-rsqrt 11006  df-abs 11007  df-clim 11286  df-proddc 11558

This theorem is referenced by:  fprodp1  11607  fprodm1s  11608