Theorem fprodeq0 14705

Description: Anything 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

Step	Hyp	Ref	Expression
1		eluzel2 11692	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> N e. ZZ )
2	1	adantl 482	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ZZ )
3	2	zred 11482	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. RR )
4	3	ltp1d 10954	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N < ( N + 1 ) )
5		fzdisj 12368	. . . 4 |- ( N < ( N + 1 ) -> ( ( M ... N ) i^i ( ( N + 1 ) ... K ) ) = (/) )
6	4, 5	syl 17	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( M ... N ) i^i ( ( N + 1 ) ... K ) ) = (/) )
7		fprodeq0.2	. . . . . . . 8 |- ( ph -> N e. Z )
8		eluzel2 11692	. . . . . . . . 9 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
9		fprodeq0.1	. . . . . . . . 9 |- Z = ( ZZ>= ` M )
10	8, 9	eleq2s 2719	. . . . . . . 8 |- ( N e. Z -> M e. ZZ )
11	7, 10	syl 17	. . . . . . 7 |- ( ph -> M e. ZZ )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> M e. ZZ )
13		eluzelz 11697	. . . . . . 7 |- ( K e. ( ZZ>= ` N ) -> K e. ZZ )
14	13	adantl 482	. . . . . 6 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> K e. ZZ )
15	12, 14, 2	3jca 1242	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) )
16		eluzle 11700	. . . . . . . 8 |- ( N e. ( ZZ>= ` M ) -> M <_ N )
17	16, 9	eleq2s 2719	. . . . . . 7 |- ( N e. Z -> M <_ N )
18	7, 17	syl 17	. . . . . 6 |- ( ph -> M <_ N )
19		eluzle 11700	. . . . . 6 |- ( K e. ( ZZ>= ` N ) -> N <_ K )
20	18, 19	anim12i 590	. . . . 5 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M <_ N /\ N <_ K ) )
21		elfz2 12333	. . . . 5 |- ( N e. ( M ... K ) <-> ( ( M e. ZZ /\ K e. ZZ /\ N e. ZZ ) /\ ( M <_ N /\ N <_ K ) ) )
22	15, 20, 21	sylanbrc 698	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> N e. ( M ... K ) )
23		fzsplit 12367	. . . 4 |- ( N e. ( M ... K ) -> ( M ... K ) = ( ( M ... N ) u. ( ( N + 1 ) ... K ) ) )
24	22, 23	syl 17	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) = ( ( M ... N ) u. ( ( N + 1 ) ... K ) ) )
25		fzfid 12772	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( M ... K ) e. Fin )
26		elfzuz 12338	. . . . . 6 |- ( k e. ( M ... K ) -> k e. ( ZZ>= ` M ) )
27	26, 9	syl6eleqr 2712	. . . . 5 |- ( k e. ( M ... K ) -> k e. Z )
28		fprodeq0.3	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. CC )
29	27, 28	sylan2 491	. . . 4 |- ( ( ph /\ k e. ( M ... K ) ) -> A e. CC )
30	29	adantlr 751	. . 3 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ k e. ( M ... K ) ) -> A e. CC )
31	6, 24, 25, 30	fprodsplit 14696	. 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 ) )
32	7, 9	syl6eleq 2711	. . . . . 6 |- ( ph -> N e. ( ZZ>= ` M ) )
33		elfzuz 12338	. . . . . . . 8 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
34	33, 9	syl6eleqr 2712	. . . . . . 7 |- ( k e. ( M ... N ) -> k e. Z )
35	34, 28	sylan2 491	. . . . . 6 |- ( ( ph /\ k e. ( M ... N ) ) -> A e. CC )
36	32, 35	fprodm1s 14700	. . . . 5 |- ( ph -> prod_ k e. ( M ... N ) A = ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) )
37		fprodeq0.4	. . . . . . 7 |- ( ( ph /\ k = N ) -> A = 0 )
38	7, 37	csbied 3560	. . . . . 6 |- ( ph -> [_ N / k ]_ A = 0 )
39	38	oveq2d 6666	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. [_ N / k ]_ A ) = ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) )
40		fzfid 12772	. . . . . . 7 |- ( ph -> ( M ... ( N - 1 ) ) e. Fin )
41		elfzuz 12338	. . . . . . . . 9 |- ( k e. ( M ... ( N - 1 ) ) -> k e. ( ZZ>= ` M ) )
42	41, 9	syl6eleqr 2712	. . . . . . . 8 |- ( k e. ( M ... ( N - 1 ) ) -> k e. Z )
43	42, 28	sylan2 491	. . . . . . 7 |- ( ( ph /\ k e. ( M ... ( N - 1 ) ) ) -> A e. CC )
44	40, 43	fprodcl 14682	. . . . . 6 |- ( ph -> prod_ k e. ( M ... ( N - 1 ) ) A e. CC )
45	44	mul01d 10235	. . . . 5 |- ( ph -> ( prod_ k e. ( M ... ( N - 1 ) ) A x. 0 ) = 0 )
46	36, 39, 45	3eqtrd 2660	. . . 4 |- ( ph -> prod_ k e. ( M ... N ) A = 0 )
47	46	adantr 481	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... N ) A = 0 )
48	47	oveq1d 6665	. 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 ) )
49		fzfid 12772	. . . 4 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( ( N + 1 ) ... K ) e. Fin )
50	9	peano2uzs 11742	. . . . . . . . 9 |- ( N e. Z -> ( N + 1 ) e. Z )
51	7, 50	syl 17	. . . . . . . 8 |- ( ph -> ( N + 1 ) e. Z )
52		elfzuz 12338	. . . . . . . 8 |- ( k e. ( ( N + 1 ) ... K ) -> k e. ( ZZ>= ` ( N + 1 ) ) )
53	9	uztrn2 11705	. . . . . . . 8 |- ( ( ( N + 1 ) e. Z /\ k e. ( ZZ>= ` ( N + 1 ) ) ) -> k e. Z )
54	51, 52, 53	syl2an 494	. . . . . . 7 |- ( ( ph /\ k e. ( ( N + 1 ) ... K ) ) -> k e. Z )
55	54	adantrl 752	. . . . . 6 |- ( ( ph /\ ( K e. ( ZZ>= ` N ) /\ k e. ( ( N + 1 ) ... K ) ) ) -> k e. Z )
56	55, 28	syldan 487	. . . . 5 |- ( ( ph /\ ( K e. ( ZZ>= ` N ) /\ k e. ( ( N + 1 ) ... K ) ) ) -> A e. CC )
57	56	anassrs 680	. . . 4 |- ( ( ( ph /\ K e. ( ZZ>= ` N ) ) /\ k e. ( ( N + 1 ) ... K ) ) -> A e. CC )
58	49, 57	fprodcl 14682	. . 3 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( ( N + 1 ) ... K ) A e. CC )
59	58	mul02d 10234	. 2 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> ( 0 x. prod_ k e. ( ( N + 1 ) ... K ) A ) = 0 )
60	31, 48, 59	3eqtrd 2660	1 |- ( ( ph /\ K e. ( ZZ>= ` N ) ) -> prod_ k e. ( M ... K ) A = 0 )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   [_csb 3533   u. cun 3572   i^i cin 3573   (/)c0 3915   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   prod_cprod 14635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-prod 14636

This theorem is referenced by:  bcc0  38539