Theorem prodfdivap 11510

Description: The quotient of two products. (Contributed by Scott Fenton, 15-Jan-2018.) (Revised by Jim Kingdon, 24-Mar-2024.)

Hypotheses

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

Assertion

Ref	Expression
prodfdivap	|- ( 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 prodfdivap

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prodfdiv.1	. . . 4 |- ( ph -> N e. ( ZZ>= ` M ) )
2		prodfdivap.3	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC )
3		elfzuz 9977	. . . . 5 |- ( k e. ( M ... N ) -> k e. ( ZZ>= ` M ) )
4		prodfdivap.4	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) # 0 )
5	3, 4	sylan2 284	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) # 0 )
6		eqid 2170	. . . . . 6 |- ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) = ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) )
7		fveq2 5496	. . . . . . 7 |- ( n = k -> ( G ` n ) = ( G ` k ) )
8	7	oveq2d 5869	. . . . . 6 |- ( n = k -> ( 1 / ( G ` n ) ) = ( 1 / ( G ` k ) ) )
9		simpr 109	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
10	2, 4	recclapd 8698	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` k ) ) e. CC )
11	6, 8, 9, 10	fvmptd3 5589	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) = ( 1 / ( G ` k ) ) )
12	3, 11	sylan2 284	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) = ( 1 / ( G ` k ) ) )
13	11, 10	eqeltrd 2247	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
14	1, 2, 5, 12, 13	prodfrecap 11509	. . 3 |- ( ph -> ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) = ( 1 / ( seq M ( x. , G ) ` N ) ) )
15	14	oveq2d 5869	. 2 |- ( ph -> ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) ) = ( ( seq M ( x. , F ) ` N ) x. ( 1 / ( seq M ( x. , G ) ` N ) ) ) )
16		prodfdivap.2	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( F ` k ) e. CC )
17		eleq1w 2231	. . . . . . . . 9 |- ( k = n -> ( k e. ( ZZ>= ` M ) <-> n e. ( ZZ>= ` M ) ) )
18	17	anbi2d 461	. . . . . . . 8 |- ( k = n -> ( ( ph /\ k e. ( ZZ>= ` M ) ) <-> ( ph /\ n e. ( ZZ>= ` M ) ) ) )
19		fveq2 5496	. . . . . . . . 9 |- ( k = n -> ( G ` k ) = ( G ` n ) )
20	19	eleq1d 2239	. . . . . . . 8 |- ( k = n -> ( ( G ` k ) e. CC <-> ( G ` n ) e. CC ) )
21	18, 20	imbi12d 233	. . . . . . 7 |- ( k = n -> ( ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) e. CC ) <-> ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) e. CC ) ) )
22	21, 2	chvarvv 1901	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) e. CC )
23	19	breq1d 3999	. . . . . . . 8 |- ( k = n -> ( ( G ` k ) # 0 <-> ( G ` n ) # 0 ) )
24	18, 23	imbi12d 233	. . . . . . 7 |- ( k = n -> ( ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) # 0 ) <-> ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 ) ) )
25	24, 4	chvarvv 1901	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 )
26	22, 25	recclapd 8698	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` n ) ) e. CC )
27	26	fmpttd 5651	. . . 4 |- ( ph -> ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) : ( ZZ>= ` M ) --> CC )
28	27	ffvelrnda 5631	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
29	16, 2, 4	divrecapd 8710	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( F ` k ) / ( G ` k ) ) = ( ( F ` k ) x. ( 1 / ( G ` k ) ) ) )
30		prodfdivap.5	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) / ( G ` k ) ) )
31	11	oveq2d 5869	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( F ` k ) x. ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) ) = ( ( F ` k ) x. ( 1 / ( G ` k ) ) ) )
32	29, 30, 31	3eqtr4d 2213	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( H ` k ) = ( ( F ` k ) x. ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) ) )
33	1, 16, 28, 32	prod3fmul 11504	. 2 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) x. ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) ) )
34		eqid 2170	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
35		eluzel2 9492	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
37	34, 36, 16	prodf 11501	. . . 4 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
38	37, 1	ffvelrnd 5632	. . 3 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
39	34, 36, 2	prodf 11501	. . . 4 |- ( ph -> seq M ( x. , G ) : ( ZZ>= ` M ) --> CC )
40	39, 1	ffvelrnd 5632	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) e. CC )
41	1, 2, 5	prodfap0 11508	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) # 0 )
42	38, 40, 41	divrecapd 8710	. 2 |- ( ph -> ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) = ( ( seq M ( x. , F ) ` N ) x. ( 1 / ( seq M ( x. , G ) ` N ) ) ) )
43	15, 33, 42	3eqtr4d 2213	1 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   class class class wbr 3989   |-> cmpt 4050   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   1c1 7775   x. cmul 7779   # cap 8500   / cdiv 8589   ZZcz 9212   ZZ>=cuz 9487   ...cfz 9965   seqcseq 10401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-uz 9488  df-fz 9966  df-fzo 10099  df-seqfrec 10402

This theorem is referenced by: (None)