Theorem fprodm1 12109

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 10300	. . . . 5 |- -. ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) )
2		fprodm1.1	. . . . . . . . 9 |- ( ph -> N e. ( ZZ>= ` M ) )
3		eluzelz 9731	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
4	2, 3	syl 14	. . . . . . . 8 |- ( ph -> N e. ZZ )
5	4	zcnd 9570	. . . . . . 7 |- ( ph -> N e. CC )
6		1cnd 8162	. . . . . . 7 |- ( ph -> 1 e. CC )
7	5, 6	npcand 8461	. . . . . 6 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
8	7	eleq1d 2298	. . . . 5 |- ( ph -> ( ( ( N - 1 ) + 1 ) e. ( M ... ( N - 1 ) ) <-> N e. ( M ... ( N - 1 ) ) ) )
9	1, 8	mtbii 678	. . . 4 |- ( ph -> -. N e. ( M ... ( N - 1 ) ) )
10		disjsn 3728	. . . 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 9727	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
13	2, 12	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
14		peano2zm 9484	. . . . . . 7 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
15	13, 14	syl 14	. . . . . 6 |- ( ph -> ( M - 1 ) e. ZZ )
16	13	zcnd 9570	. . . . . . . . 9 |- ( ph -> M e. CC )
17	16, 6	npcand 8461	. . . . . . . 8 |- ( ph -> ( ( M - 1 ) + 1 ) = M )
18	17	fveq2d 5631	. . . . . . 7 |- ( ph -> ( ZZ>= ` ( ( M - 1 ) + 1 ) ) = ( ZZ>= ` M ) )
19	2, 18	eleqtrrd 2309	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` ( ( M - 1 ) + 1 ) ) )
20		eluzp1m1 9746	. . . . . 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 10275	. . . . 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 6017	. . . 4 |- ( ph -> ( M ... ( ( N - 1 ) + 1 ) ) = ( M ... N ) )
25	7	sneqd 3679	. . . . 5 |- ( ph -> { ( ( N - 1 ) + 1 ) } = { N } )
26	25	uneq2d 3358	. . . 4 |- ( ph -> ( ( M ... ( N - 1 ) ) u. { ( ( N - 1 ) + 1 ) } ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
27	23, 24, 26	3eqtr3d 2270	. . 3 |- ( ph -> ( M ... N ) = ( ( M ... ( N - 1 ) ) u. { N } ) )
28	13, 4	fzfigd 10653	. . 3 |- ( ph -> ( M ... N ) e. Fin )
29		elfzelz 10221	. . . . . 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 9484	. . . . . 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 10236	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ ( N - 1 ) e. ZZ ) -> DECID j e. ( M ... ( N - 1 ) ) )
36	30, 31, 34, 35	syl3anc 1271	. . . 4 |- ( ( ph /\ j e. ( M ... N ) ) -> DECID j e. ( M ... ( N - 1 ) ) )
37	36	ralrimiva 2603	. . 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 12107	. 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 2298	. . . . 5 |- ( k = N -> ( A e. CC <-> B e. CC ) )
42	38	ralrimiva 2603	. . . . 5 |- ( ph -> A. k e. ( M ... N ) A e. CC )
43		eluzfz2 10228	. . . . . 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 2912	. . . 4 |- ( ph -> B e. CC )
46	40	prodsn 12104	. . . 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 6017	. 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 2262	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 839   = wceq 1395   e. wcel 2200   u. cun 3195   i^i cin 3196   (/)c0 3491   {csn 3666   ` cfv 5318  (class class class)co 6001   CCcc 7997   1c1 8000   + caddc 8002   x. cmul 8004   - cmin 8317   ZZcz 9446   ZZ>=cuz 9722   ...cfz 10204   prod_cprod 12061

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-mulrcl 8098  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-precex 8109  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115  ax-pre-mulgt0 8116  ax-pre-mulext 8117  ax-arch 8118  ax-caucvg 8119

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-isom 5327  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-irdg 6516  df-frec 6537  df-1o 6562  df-oadd 6566  df-er 6680  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-reap 8722  df-ap 8729  df-div 8820  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-n0 9370  df-z 9447  df-uz 9723  df-q 9815  df-rp 9850  df-fz 10205  df-fzo 10339  df-seqfrec 10670  df-exp 10761  df-ihash 10998  df-cj 11353  df-re 11354  df-im 11355  df-rsqrt 11509  df-abs 11510  df-clim 11790  df-proddc 12062

This theorem is referenced by:  fprodp1  12111  fprodm1s  12112