Theorem fprodeq0 12043

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 9688	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> N e. ZZ )
2	1	adantl 277	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ZZ )
3	2	zred 9530	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. RR )
4	3	ltp1d 9038	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N < ( N + 1 ) )
5		fzdisj 10209	. . . 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 9688	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9		fprodeq0.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
10	8, 9	eleq2s 2302	. . . . . . . 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 9692	. . . . . . 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 1180	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) )
16		eluzle 9695	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
17	16, 9	eleq2s 2302	. . . . . . 7 |- ( N e. Z -> M <_ N )
18	7, 17	syl 14	. . . . . 6 |- ( ph -> M <_ N )
19		eluzle 9695	. . . . . 6 |- ( K e. ( ZZ>= ` N ) -> N <_ K )
20	18, 19	anim12i 338	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M <_ N /\ N <_ K ) )
21		elfz2 10172	. . . . 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 10208	. . . 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 10613	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) e. Fin )
26		elfzelz 10182	. . . . . 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 10197	. . . . 5 |- ( ( j e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> DECID j e. ( M ... N ) )
31	27, 28, 29, 30	syl3anc 1250	. . . 4 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ j e. ( M ... K ) ) -> DECID j e. ( M ... N ) )
32	31	ralrimiva 2581	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> A. j e. ( M ... K )DECID j e. ( M ... N ) )
33		elfzuz 10178	. . . . . 6 |- ( k e. ( M ... K ) -> k e. ( ZZ>= ` M ) )
34	33, 9	eleqtrrdi 2301	. . . . 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 12022	. 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 2300	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
40		elfzuz 10178	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
41	40, 9	eleqtrrdi 2301	. . . . . . 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 12027	. . . . 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 3148	. . . . . 6 |- ( ph -> [_ N / k ]_ A = 0 )
46	45	oveq2d 5983	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) = ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) )
47		eluzelz 9692	. . . . . . . . . 10 |- ( N e. ( ZZ>= ` M ) -> N e. ZZ )
48	39, 47	syl 14	. . . . . . . . 9 |- ( ph -> N e. ZZ )
49		peano2zm 9445	. . . . . . . . 9 |- ( N e. ZZ -> ( N - 1 ) e. ZZ )
50	48, 49	syl 14	. . . . . . . 8 |- ( ph -> ( N - 1 ) e. ZZ )
51	11, 50	fzfigd 10613	. . . . . . 7 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
52		elfzuz 10178	. . . . . . . . 9 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
53	52, 9	eleqtrrdi 2301	. . . . . . . 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 12033	. . . . . 6 |- ( ph -> prod_ k e. ( M ... ( N - 1 ) ) A e. CC )
56	55	mul01d 8500	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) = 0 )
57	43, 46, 56	3eqtrd 2244	. . . 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 5982	. 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 9533	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( N + 1 ) e. ZZ )
61	60, 14	fzfigd 10613	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( N + 1 ) ... K ) e. Fin )
62	9	peano2uzs 9740	. . . . . . . . 9 |- ( N e. Z -> ( N + 1 ) e. Z )
63	7, 62	syl 14	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. Z )
64		elfzuz 10178	. . . . . . . 8 |- ( k e. ( ( N + 1 ) ... K ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
65	9	uztrn2 9701	. . . . . . . 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 12033	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( ( N + 1 ) ... K ) A e. CC )
71	70	mul02d 8499	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( 0 x. prod_ k e. ( ( N + 1 ) ... K ) A ) = 0 )
72	38, 59, 71	3eqtrd 2244	1 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2178   [_csb 3101   u. cun 3172   i^i cin 3173   (/)c0 3468   class class class wbr 4059   ` cfv 5290  (class class class)co 5967   CCcc 7958   0cc0 7960   1c1 7961   + caddc 7963   x. cmul 7965   < clt 8142   <_ cle 8143   - cmin 8278   ZZcz 9407   ZZ>=cuz 9683   ...cfz 10165   prod_cprod 11976

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-frec 6500  df-1o 6525  df-oadd 6529  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-seqfrec 10630  df-exp 10721  df-ihash 10958  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-clim 11705  df-proddc 11977

This theorem is referenced by: (None)