Theorem fprodm1 11499

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 10007	. . . . 5 |- -. ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) )
2		fprodm1.1	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9449	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
5	4	zcnd 9288	. . . . . . 7 |- ( ph -> N e. CC )
6		1cnd 7895	. . . . . . 7 |- ( ph -> 1 e. CC )
7	5, 6	npcand 8191	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
8	7	eleq1d 2226	. . . . 5 |- ( ph -> ( ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) ) <-> N e. ( M ... ( N - 1 ) ) ) )
9	1, 8	mtbii 664	. . . 4 |- ( ph -> -. N e. ( M ... ( N - 1 ) ) )
10		disjsn 3622	. . . 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 9445	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	2, 12	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
14		peano2zm 9206	. . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . 6 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9288	. . . . . . . . 9 |- ( ph -> M e. CC )
17	16, 6	npcand 8191	. . . . . . . 8 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
18	17	fveq2d 5473	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
19	2, 18	eleqtrrd 2237	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
20		eluzp1m1 9463	. . . . . 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 9982	. . . . 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 5841	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
25	7	sneqd 3573	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
26	25	uneq2d 3261	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
27	23, 24, 26	3eqtr3d 2198	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
28	13, 4	fzfigd 10334	. . 3 |- ( ph -> ( M ... N ) e. Fin )
29		elfzelz 9929	. . . . . 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 9206	. . . . . 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 9943	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ ( N - 1 ) e. ZZ ) -> DECID j e. ( M ... ( N - 1 ) ) )
36	30, 31, 34, 35	syl3anc 1220	. . . 4 |- ( ( ph /\ j e. ( M ... N ) ) -> DECID j e. ( M ... ( N - 1 ) ) )
37	36	ralrimiva 2530	. . 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 11497	. 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 2226	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
42	38	ralrimiva 2530	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
43		eluzfz2 9935	. . . . . 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 2821	. . . 4 |- ( ph -> B e. CC )
46	40	prodsn 11494	. . . 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 5841	. 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 2190	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 820   = wceq 1335   e. wcel 2128   u. cun 3100   i^i cin 3101   (/)c0 3394   {csn 3560   ` cfv 5171  (class class class)co 5825   CCcc 7731   1c1 7734   + caddc 7736   x. cmul 7738   - cmin 8047   ZZcz 9168   ZZ>=cuz 9440   ...cfz 9913   prod_cprod 11451

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-nul 4091  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497  ax-iinf 4548  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-mulrcl 7832  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-0lt1 7839  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-precex 7843  ax-cnre 7844  ax-pre-ltirr 7845  ax-pre-ltwlin 7846  ax-pre-lttrn 7847  ax-pre-apti 7848  ax-pre-ltadd 7849  ax-pre-mulgt0 7850  ax-pre-mulext 7851  ax-arch 7852  ax-caucvg 7853

This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rmo 2443  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-if 3506  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-tr 4064  df-id 4254  df-po 4257  df-iso 4258  df-iord 4327  df-on 4329  df-ilim 4330  df-suc 4332  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-isom 5180  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-recs 6253  df-irdg 6318  df-frec 6339  df-1o 6364  df-oadd 6368  df-er 6481  df-en 6687  df-dom 6688  df-fin 6689  df-pnf 7915  df-mnf 7916  df-xr 7917  df-ltxr 7918  df-le 7919  df-sub 8049  df-neg 8050  df-reap 8451  df-ap 8458  df-div 8547  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-n0 9092  df-z 9169  df-uz 9441  df-q 9530  df-rp 9562  df-fz 9914  df-fzo 10046  df-seqfrec 10349  df-exp 10423  df-ihash 10654  df-cj 10746  df-re 10747  df-im 10748  df-rsqrt 10902  df-abs 10903  df-clim 11180  df-proddc 11452

This theorem is referenced by:  fprodp1  11501  fprodm1s  11502