Theorem fprodm1 12284

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 10438	. . . . 5 |- -. ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) )
2		fprodm1.1	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9863	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
5	4	zcnd 9701	. . . . . . 7 |- ( ph -> N e. CC )
6		1cnd 8290	. . . . . . 7 |- ( ph -> 1 e. CC )
7	5, 6	npcand 8588	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
8	7	eleq1d 2301	. . . . 5 |- ( ph -> ( ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) ) <-> N e. ( M ... ( N - 1 ) ) ) )
9	1, 8	mtbii 681	. . . 4 |- ( ph -> -. N e. ( M ... ( N - 1 ) ) )
10		disjsn 3751	. . . 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 9858	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	2, 12	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
14		peano2zm 9615	. . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . 6 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9701	. . . . . . . . 9 |- ( ph -> M e. CC )
17	16, 6	npcand 8588	. . . . . . . 8 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
18	17	fveq2d 5674	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
19	2, 18	eleqtrrd 2312	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
20		eluzp1m1 9878	. . . . . 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 10413	. . . . 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 6066	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
25	7	sneqd 3702	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
26	25	uneq2d 3373	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
27	23, 24, 26	3eqtr3d 2273	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
28	13, 4	fzfigd 10793	. . 3 |- ( ph -> ( M ... N ) e. Fin )
29		elfzelz 10359	. . . . . 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 9615	. . . . . 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 10374	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ ( N - 1 ) e. ZZ ) -> DECID j e. ( M ... ( N - 1 ) ) )
36	30, 31, 34, 35	syl3anc 1274	. . . 4 |- ( ( ph /\ j e. ( M ... N ) ) -> DECID j e. ( M ... ( N - 1 ) ) )
37	36	ralrimiva 2615	. . 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 12282	. 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 2301	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
42	38	ralrimiva 2615	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
43		eluzfz2 10366	. . . . . 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 2926	. . . 4 |- ( ph -> B e. CC )
46	40	prodsn 12279	. . . 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 6066	. 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 2265	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 842   = wceq 1398   e. wcel 2203   u. cun 3209   i^i cin 3210   (/)c0 3508   {csn 3689   ` cfv 5352  (class class class)co 6050   CCcc 8125   1c1 8128   + caddc 8130   x. cmul 8132   - cmin 8444   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342   prod_cprod 12236

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-proddc 12237

This theorem is referenced by:  fprodp1  12286  fprodm1s  12287