Theorem fprodm1 11561

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 10060	. . . . 5 |- -. ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) )
2		fprodm1.1	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9496	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
5	4	zcnd 9335	. . . . . . 7 |- ( ph -> N e. CC )
6		1cnd 7936	. . . . . . 7 |- ( ph -> 1 e. CC )
7	5, 6	npcand 8234	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
8	7	eleq1d 2239	. . . . 5 |- ( ph -> ( ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) ) <-> N e. ( M ... ( N - 1 ) ) ) )
9	1, 8	mtbii 669	. . . 4 |- ( ph -> -. N e. ( M ... ( N - 1 ) ) )
10		disjsn 3645	. . . 4 |- ( ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) <-> -. N e. ( M ... ( N - 1 ) ) )
11	9, 10	sylibr 133	. . 3 |- ( ph -> ( ( M ... ( N - 1 ) ) i^i { N } ) = (/) )
12		eluzel2 9492	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	2, 12	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
14		peano2zm 9250	. . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . 6 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9335	. . . . . . . . 9 |- ( ph -> M e. CC )
17	16, 6	npcand 8234	. . . . . . . 8 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
18	17	fveq2d 5500	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
19	2, 18	eleqtrrd 2250	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
20		eluzp1m1 9510	. . . . . 6 |- ( ( ( M - 1 ) e. ZZ /\ N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) ) -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
21	15, 19, 20	syl2anc 409	. . . . 5 |- ( ph -> ( N - 1 ) e. ( ZZ>= ` ( M - 1 ) ) )
22		fzsuc2 10035	. . . . 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 409	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) )
24	7	oveq2d 5869	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
25	7	sneqd 3596	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
26	25	uneq2d 3281	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
27	23, 24, 26	3eqtr3d 2211	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
28	13, 4	fzfigd 10387	. . 3 |- ( ph -> ( M ... N ) e. Fin )
29		elfzelz 9981	. . . . . 6 |- ( j e. ( M ... N ) -> j e. ZZ )
30	29	adantl 275	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> j e. ZZ )
31	13	adantr 274	. . . . 5 |- ( ( ph /\ j e. ( M ... N ) ) -> M e. ZZ )
32	4	adantr 274	. . . . . 6 |- ( ( ph /\ j e. ( M ... N ) ) -> N e. ZZ )
33		peano2zm 9250	. . . . . 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 9996	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ ( N - 1 ) e. ZZ ) -> DECID j e. ( M ... ( N - 1 ) ) )
36	30, 31, 34, 35	syl3anc 1233	. . . 4 |- ( ( ph /\ j e. ( M ... N ) ) -> DECID j e. ( M ... ( N - 1 ) ) )
37	36	ralrimiva 2543	. . 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 11559	. 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 2239	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
42	38	ralrimiva 2543	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
43		eluzfz2 9988	. . . . . 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 2839	. . . 4 |- ( ph -> B e. CC )
46	40	prodsn 11556	. . . 4 |- ( ( N e. ( ZZ>= ` M ) /\ B e. CC ) -> prod_ k e. { N } A = B )
47	2, 45, 46	syl2anc 409	. . 3 |- ( ph -> prod_ k e. { N } A = B )
48	47	oveq2d 5869	. 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 2203	1 |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103  DECID wdc 829   = wceq 1348   e. wcel 2141   u. cun 3119   i^i cin 3120   (/)c0 3414   {csn 3583   ` cfv 5198  (class class class)co 5853   CCcc 7772   1c1 7775   + caddc 7777   x. cmul 7779   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   prod_cprod 11513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-proddc 11514

This theorem is referenced by:  fprodp1  11563  fprodm1s  11564