Theorem fprodeq0 11609

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 9522	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> N e. ZZ )
2	1	adantl 277	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ZZ )
3	2	zred 9364	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. RR )
4	3	ltp1d 8876	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N < ( N + 1 ) )
5		fzdisj 10038	. . . 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 9522	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9		fprodeq0.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
10	8, 9	eleq2s 2272	. . . . . . . 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 9526	. . . . . . 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 1177	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) )
16		eluzle 9529	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
17	16, 9	eleq2s 2272	. . . . . . 7 |- ( N e. Z -> M <_ N )
18	7, 17	syl 14	. . . . . 6 |- ( ph -> M <_ N )
19		eluzle 9529	. . . . . 6 |- ( K e. ( ZZ>= ` N ) -> N <_ K )
20	18, 19	anim12i 338	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M <_ N /\ N <_ K ) )
21		elfz2 10002	. . . . 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 10037	. . . 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 10417	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) e. Fin )
26		elfzelz 10011	. . . . . 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 10026	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID j e. ( M ... N ) )
31	27, 28, 29, 30	syl3anc 1238	. . . 4 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> DECID j e. ( M ... N ) )
32	31	ralrimiva 2550	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> A. j e. ( M ... K )DECID j e. ( M ... N ) )
33		elfzuz 10007	. . . . . 6 |- ( k e. ( M ... K ) -> k e. ( ZZ>= ` M ) )
34	33, 9	eleqtrrdi 2271	. . . . 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 11588	. 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 2270	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
40		elfzuz 10007	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
41	40, 9	eleqtrrdi 2271	. . . . . . 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 11593	. . . . 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 3103	. . . . . 6 |- ( ph -> [_ N / k ]_ A = 0 )
46	45	oveq2d 5885	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) = ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) )
47		eluzelz 9526	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
48	39, 47	syl 14	. . . . . . . . 9 |- ( ph -> N e. ZZ )
49		peano2zm 9280	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
50	48, 49	syl 14	. . . . . . . 8 |- ( ph -> ( N - 1 ) e. ZZ )
51	11, 50	fzfigd 10417	. . . . . . 7 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
52		elfzuz 10007	. . . . . . . . 9 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
53	52, 9	eleqtrrdi 2271	. . . . . . . 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 11599	. . . . . 6 |- ( ph -> prod_ k e. ( M ... ( N - 1 ) ) A e. CC )
56	55	mul01d 8340	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) = 0 )
57	43, 46, 56	3eqtrd 2214	. . . 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 5884	. 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 9367	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( N + 1 ) e. ZZ )
61	60, 14	fzfigd 10417	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( N + 1 ) ... K ) e. Fin )
62	9	peano2uzs 9573	. . . . . . . . 9 |- ( N e. Z -> ( N + 1 ) e. Z )
63	7, 62	syl 14	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. Z )
64		elfzuz 10007	. . . . . . . 8 |- ( k e. ( ( N + 1 ) ... K ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
65	9	uztrn2 9534	. . . . . . . 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 11599	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( ( N + 1 ) ... K ) A e. CC )
71	70	mul02d 8339	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( 0 x. prod_ k e. ( ( N + 1 ) ... K ) A ) = 0 )
72	38, 59, 71	3eqtrd 2214	1 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 834   /\ w3a 978   = wceq 1353   e. wcel 2148   [_csb 3057   u. cun 3127   i^i cin 3128   (/)c0 3422   class class class wbr 4000   ` cfv 5212  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   < clt 7982   <_ cle 7983   - cmin 8118   ZZcz 9242   ZZ>=cuz 9517   ...cfz 9995   prod_cprod 11542

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543

This theorem is referenced by: (None)