Theorem fprodeq0 12328

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 9876	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> N e. ZZ )
2	1	adantl 277	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ZZ )
3	2	zred 9718	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. RR )
4	3	ltp1d 9221	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N < ( N + 1 ) )
5		fzdisj 10406	. . . 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 9876	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9		fprodeq0.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
10	8, 9	eleq2s 2329	. . . . . . . 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 9881	. . . . . . 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 1204	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) )
16		eluzle 9884	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
17	16, 9	eleq2s 2329	. . . . . . 7 |- ( N e. Z -> M <_ N )
18	7, 17	syl 14	. . . . . 6 |- ( ph -> M <_ N )
19		eluzle 9884	. . . . . 6 |- ( K e. ( ZZ>= ` N ) -> N <_ K )
20	18, 19	anim12i 338	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M <_ N /\ N <_ K ) )
21		elfz2 10368	. . . . 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 10405	. . . 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 10817	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) e. Fin )
26		elfzelz 10378	. . . . . 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 10394	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID j e. ( M ... N ) )
31	27, 28, 29, 30	syl3anc 1274	. . . 4 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> DECID j e. ( M ... N ) )
32	31	ralrimiva 2617	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> A. j e. ( M ... K )DECID j e. ( M ... N ) )
33		elfzuz 10374	. . . . . 6 |- ( k e. ( M ... K ) -> k e. ( ZZ>= ` M ) )
34	33, 9	eleqtrrdi 2328	. . . . 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 12307	. 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 2327	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
40		elfzuz 10374	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
41	40, 9	eleqtrrdi 2328	. . . . . . 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 12312	. . . . 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 3188	. . . . . 6 |- ( ph -> [_ N / k ]_ A = 0 )
46	45	oveq2d 6074	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) = ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) )
47		eluzelz 9881	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
48	39, 47	syl 14	. . . . . . . . 9 |- ( ph -> N e. ZZ )
49		peano2zm 9632	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
50	48, 49	syl 14	. . . . . . . 8 |- ( ph -> ( N - 1 ) e. ZZ )
51	11, 50	fzfigd 10817	. . . . . . 7 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
52		elfzuz 10374	. . . . . . . . 9 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
53	52, 9	eleqtrrdi 2328	. . . . . . . 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 12318	. . . . . 6 |- ( ph -> prod_ k e. ( M ... ( N - 1 ) ) A e. CC )
56	55	mul01d 8683	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) = 0 )
57	43, 46, 56	3eqtrd 2271	. . . 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 6073	. 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 9721	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( N + 1 ) e. ZZ )
61	60, 14	fzfigd 10817	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( N + 1 ) ... K ) e. Fin )
62	9	peano2uzs 9934	. . . . . . . . 9 |- ( N e. Z -> ( N + 1 ) e. Z )
63	7, 62	syl 14	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. Z )
64		elfzuz 10374	. . . . . . . 8 |- ( k e. ( ( N + 1 ) ... K ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
65	9	uztrn2 9890	. . . . . . . 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 12318	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( ( N + 1 ) ... K ) A e. CC )
71	70	mul02d 8682	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( 0 x. prod_ k e. ( ( N + 1 ) ... K ) A ) = 0 )
72	38, 59, 71	3eqtrd 2271	1 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   [_csb 3141   u. cun 3212   i^i cin 3213   (/)c0 3512   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   prod_cprod 12261

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262  ax-caucvg 8263

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-isom 5366  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-uz 9872  df-q 9970  df-rp 10005  df-fz 10362  df-fzo 10499  df-seqfrec 10834  df-exp 10925  df-ihash 11164  df-cj 11552  df-re 11553  df-im 11554  df-rsqrt 11708  df-abs 11709  df-clim 11989  df-proddc 12262

This theorem is referenced by: (None)