Theorem fprodeq0 12149

Description: Any finite product containing a zero term is itself zero. (Contributed by Scott Fenton, 27-Dec-2017.)

Hypotheses

Ref	Expression
fprodeq0.1	|- Z = ( ZZ>= ` M )
fprodeq0.2	|- ( ph -> N e. Z )
fprodeq0.3	|- ( ( ph /\ k e. Z ) -> A e. CC )
fprodeq0.4	|- ( ( ph /\ k = N ) -> A = 0 )

Assertion

Ref	Expression
fprodeq0	|- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )

Distinct variable groups:   k, K   k, M   k, N   k, Z   ph, k

Allowed substitution hint:   A( k)

Proof of Theorem fprodeq0

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluzel2 9743	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> N e. ZZ )
2	1	adantl 277	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ZZ )
3	2	zred 9585	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. RR )
4	3	ltp1d 9093	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N < ( N + 1 ) )
5		fzdisj 10265	. . . 4 |- ( N < ( N + 1 ) -> ( ( M ... N ) i^i ( ( N + 1 ) ... K ) ) = (/) )
6	4, 5	syl 14	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( M ... N ) i^i ( ( N + 1 ) ... K ) ) = (/) )
7		fprodeq0.2	. . . . . . . 8 |- ( ph -> N e. Z )
8		eluzel2 9743	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9		fprodeq0.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
10	8, 9	eleq2s 2324	. . . . . . . 8 |- ( N e. Z -> M e. ZZ )
11	7, 10	syl 14	. . . . . . 7 |- ( ph -> M e. ZZ )
12	11	adantr 276	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> M e. ZZ )
13		eluzelz 9748	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> K e. ZZ )
14	13	adantl 277	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> K e. ZZ )
15	12, 14, 2	3jca 1201	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) )
16		eluzle 9751	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
17	16, 9	eleq2s 2324	. . . . . . 7 |- ( N e. Z -> M <_ N )
18	7, 17	syl 14	. . . . . 6 |- ( ph -> M <_ N )
19		eluzle 9751	. . . . . 6 |- ( K e. ( ZZ>= ` N ) -> N <_ K )
20	18, 19	anim12i 338	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M <_ N /\ N <_ K ) )
21		elfz2 10228	. . . . 5 |- ( N e. ( M ... K ) <-> ( ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) /\ ( M <_ N /\ N <_ K ) ) )
22	15, 20, 21	sylanbrc 417	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ( M ... K ) )
23		fzsplit 10264	. . . 4 |- ( N e. ( M ... K ) -> ( M ... K ) = ( ( M ... N ) u. ( ( N + 1 ) ... K ) ) )
24	22, 23	syl 14	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) = ( ( M ... N ) u. ( ( N + 1 ) ... K ) ) )
25	12, 14	fzfigd 10670	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) e. Fin )
26		elfzelz 10238	. . . . . 6 |- ( j e. ( M ... K ) -> j e. ZZ )
27	26	adantl 277	. . . . 5 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> j e. ZZ )
28	12	adantr 276	. . . . 5 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> M e. ZZ )
29	2	adantr 276	. . . . 5 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> N e. ZZ )
30		fzdcel 10253	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID j e. ( M ... N ) )
31	27, 28, 29, 30	syl3anc 1271	. . . 4 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> DECID j e. ( M ... N ) )
32	31	ralrimiva 2603	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> A. j e. ( M ... K )DECID j e. ( M ... N ) )
33		elfzuz 10234	. . . . . 6 |- ( k e. ( M ... K ) -> k e. ( ZZ>= ` M ) )
34	33, 9	eleqtrrdi 2323	. . . . 5 |- ( k e. ( M ... K ) -> k e. Z )
35		fprodeq0.3	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. CC )
36	34, 35	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( M ... K ) ) -> A e. CC )
37	36	adantlr 477	. . 3 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ k e. ( M ... K ) ) -> A e. CC )
38	6, 24, 25, 32, 37	fprodsplitdc 12128	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = ( prod_ k e. ( M ... N ) A x. prod_ k e. ( ( N + 1 ) ... K ) A ) )
39	7, 9	eleqtrdi 2322	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
40		elfzuz 10234	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
41	40, 9	eleqtrrdi 2323	. . . . . . 7 |- ( k e. ( M ... N ) -> k e. Z )
42	41, 35	sylan2 286	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
43	39, 42	fprodm1s 12133	. . . . 5 |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) )
44		fprodeq0.4	. . . . . . 7 |- ( ( ph /\ k = N ) -> A = 0 )
45	7, 44	csbied 3171	. . . . . 6 |- ( ph -> [_ N / k ]_ A = 0 )
46	45	oveq2d 6026	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) = ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) )
47		eluzelz 9748	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
48	39, 47	syl 14	. . . . . . . . 9 |- ( ph -> N e. ZZ )
49		peano2zm 9500	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
50	48, 49	syl 14	. . . . . . . 8 |- ( ph -> ( N - 1 ) e. ZZ )
51	11, 50	fzfigd 10670	. . . . . . 7 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
52		elfzuz 10234	. . . . . . . . 9 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
53	52, 9	eleqtrrdi 2323	. . . . . . . 8 |- ( k e. ( M ... ( N - 1 ) ) -> k e. Z )
54	53, 35	sylan2 286	. . . . . . 7 |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> A e. CC )
55	51, 54	fprodcl 12139	. . . . . 6 |- ( ph -> prod_ k e. ( M ... ( N - 1 ) ) A e. CC )
56	55	mul01d 8555	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) = 0 )
57	43, 46, 56	3eqtrd 2266	. . . 4 |- ( ph -> prod_ k e. ( M ... N ) A = 0 )
58	57	adantr 276	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... N ) A = 0 )
59	58	oveq1d 6025	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( prod_ k e. ( M ... N ) A x. prod_ k e. ( ( N + 1 ) ... K ) A ) = ( 0 x. prod_ k e. ( ( N + 1 ) ... K ) A ) )
60	2	peano2zd 9588	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( N + 1 ) e. ZZ )
61	60, 14	fzfigd 10670	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( N + 1 ) ... K ) e. Fin )
62	9	peano2uzs 9796	. . . . . . . . 9 |- ( N e. Z -> ( N + 1 ) e. Z )
63	7, 62	syl 14	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. Z )
64		elfzuz 10234	. . . . . . . 8 |- ( k e. ( ( N + 1 ) ... K ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
65	9	uztrn2 9757	. . . . . . . 8 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
66	63, 64, 65	syl2an 289	. . . . . . 7 |- ( ( ph /\ k e. ( ( N + 1 ) ... K ) ) -> k e. Z )
67	66	adantrl 478	. . . . . 6 |- ( ( ph /\ ( K e. ( ZZ>= ` N ) /\ k e. ( ( N + 1 ) ... K ) ) ) -> k e. Z )
68	67, 35	syldan 282	. . . . 5 |- ( ( ph /\ ( K e. ( ZZ>= ` N ) /\ k e. ( ( N + 1 ) ... K ) ) ) -> A e. CC )
69	68	anassrs 400	. . . 4 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ k e. ( ( N + 1 ) ... K ) ) -> A e. CC )
70	61, 69	fprodcl 12139	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( ( N + 1 ) ... K ) A e. CC )
71	70	mul02d 8554	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( 0 x. prod_ k e. ( ( N + 1 ) ... K ) A ) = 0 )
72	38, 59, 71	3eqtrd 2266	1 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 839   /\ w3a 1002   = wceq 1395   e. wcel 2200   [_csb 3124   u. cun 3195   i^i cin 3196   (/)c0 3491   class class class wbr 4083   ` cfv 5321  (class class class)co 6010   CCcc 8013   0cc0 8015   1c1 8016   + caddc 8018   x. cmul 8020   < clt 8197   <_ cle 8198   - cmin 8333   ZZcz 9462   ZZ>=cuz 9738   ...cfz 10221   prod_cprod 12082

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 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-iinf 4681  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133  ax-arch 8134  ax-caucvg 8135

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 4385  df-po 4388  df-iso 4389  df-iord 4458  df-on 4460  df-ilim 4461  df-suc 4463  df-iom 4684  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-isom 5330  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-recs 6462  df-irdg 6527  df-frec 6548  df-1o 6573  df-oadd 6577  df-er 6693  df-en 6901  df-dom 6902  df-fin 6903  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-n0 9386  df-z 9463  df-uz 9739  df-q 9832  df-rp 9867  df-fz 10222  df-fzo 10356  df-seqfrec 10687  df-exp 10778  df-ihash 11015  df-cj 11374  df-re 11375  df-im 11376  df-rsqrt 11530  df-abs 11531  df-clim 11811  df-proddc 12083

This theorem is referenced by: (None)