Theorem prodfdiv 14628

Description: The quotient of two infinite products. (Contributed by Scott Fenton, 15-Jan-2018.)

Hypotheses

Ref	Expression
prodfdiv.1	|- ( ph -> N e. ( ZZ>= ` M ) )
prodfdiv.2	|- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
prodfdiv.3	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
prodfdiv.4	|- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) =/= 0 )
prodfdiv.5	|- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) / ( G ` k ) ) )

Assertion

Ref	Expression
prodfdiv	|- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )

Distinct variable groups:   k, F   k, G   k, H   ph, k   k, M   k, N

Proof of Theorem prodfdiv

Dummy variables x n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfdiv.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		prodfdiv.3	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC )
3		prodfdiv.4	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) =/= 0 )
4		fveq2 6191	. . . . . . 7 |- ( n = k -> ( G ` n ) = ( G ` k ) )
5	4	oveq2d 6666	. . . . . 6 |- ( n = k -> ( 1 / ( G ` n ) ) = ( 1 / ( G ` k ) ) )
6		eqid 2622	. . . . . 6 |- ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) = ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) )
7		ovex 6678	. . . . . 6 |- ( 1 / ( G ` k ) ) e. _V
8	5, 6, 7	fvmpt 6282	. . . . 5 |- ( k e. ( M ... N ) -> ( ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ` k ) = ( 1 / ( G ` k ) ) )
9	8	adantl 482	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ` k ) = ( 1 / ( G ` k ) ) )
10	1, 2, 3, 9	prodfrec 14627	. . 3 |- ( ph -> ( seq M ( x. , ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) = ( 1 / ( seq M ( x. , G ) ` N ) ) )
11	10	oveq2d 6666	. 2 |- ( ph -> ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) ) = ( ( seq M ( x. , F ) ` N ) x. ( 1 / ( seq M ( x. , G ) ` N ) ) ) )
12		prodfdiv.2	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( F ` k ) e. CC )
13		eleq1 2689	. . . . . . . . 9 |- ( k = n -> ( k e. ( M ... N ) <-> n e. ( M ... N ) ) )
14	13	anbi2d 740	. . . . . . . 8 |- ( k = n -> ( ( ph /\ k e. ( M ... N ) ) <-> ( ph /\ n e. ( M ... N ) ) ) )
15		fveq2 6191	. . . . . . . . 9 |- ( k = n -> ( G ` k ) = ( G ` n ) )
16	15	eleq1d 2686	. . . . . . . 8 |- ( k = n -> ( ( G ` k ) e. CC <-> ( G ` n ) e. CC ) )
17	14, 16	imbi12d 334	. . . . . . 7 |- ( k = n -> ( ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) e. CC ) <-> ( ( ph /\ n e. ( M ... N ) ) -> ( G ` n ) e. CC ) ) )
18	17, 2	chvarv 2263	. . . . . 6 |- ( ( ph /\ n e. ( M ... N ) ) -> ( G ` n ) e. CC )
19	15	neeq1d 2853	. . . . . . . 8 |- ( k = n -> ( ( G ` k ) =/= 0 <-> ( G ` n ) =/= 0 ) )
20	14, 19	imbi12d 334	. . . . . . 7 |- ( k = n -> ( ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) =/= 0 ) <-> ( ( ph /\ n e. ( M ... N ) ) -> ( G ` n ) =/= 0 ) ) )
21	20, 3	chvarv 2263	. . . . . 6 |- ( ( ph /\ n e. ( M ... N ) ) -> ( G ` n ) =/= 0 )
22	18, 21	reccld 10794	. . . . 5 |- ( ( ph /\ n e. ( M ... N ) ) -> ( 1 / ( G ` n ) ) e. CC )
23	22, 6	fmptd 6385	. . . 4 |- ( ph -> ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) : ( M ... N ) --> CC )
24	23	ffvelrnda 6359	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
25	12, 2, 3	divrecd 10804	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( F ` k ) / ( G ` k ) ) = ( ( F ` k ) x. ( 1 / ( G ` k ) ) ) )
26		prodfdiv.5	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) / ( G ` k ) ) )
27	9	oveq2d 6666	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( F ` k ) x. ( ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ` k ) ) = ( ( F ` k ) x. ( 1 / ( G ` k ) ) ) )
28	25, 26, 27	3eqtr4d 2666	. . 3 |- ( ( ph /\ k e. ( M ... N ) ) -> ( H ` k ) = ( ( F ` k ) x. ( ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ` k ) ) )
29	1, 12, 24, 28	prodfmul 14622	. 2 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , ( n e. ( M ... N ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) ) )
30		mulcl 10020	. . . . 5 |- ( ( k e. CC /\ x e. CC ) -> ( k x. x ) e. CC )
31	30	adantl 482	. . . 4 |- ( ( ph /\ ( k e. CC /\ x e. CC ) ) -> ( k x. x ) e. CC )
32	1, 12, 31	seqcl 12821	. . 3 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
33	1, 2, 31	seqcl 12821	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) e. CC )
34	1, 2, 3	prodfn0 14626	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) =/= 0 )
35	32, 33, 34	divrecd 10804	. 2 |- ( ph -> ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) = ( ( seq M ( x. , F ) ` N ) x. ( 1 / ( seq M ( x. , G ) ` N ) ) ) )
36	11, 29, 35	3eqtr4d 2666	1 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   / cdiv 10684   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802

This theorem is referenced by:  fproddiv  14691