Theorem prodfdivap 11712

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 10096	. . . . 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 286	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( G ` k ) # 0 )
6		eqid 2196	. . . . . 6 |- ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) = ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) )
7		fveq2 5558	. . . . . . 7 |- ( n = k -> ( G ` n ) = ( G ` k ) )
8	7	oveq2d 5938	. . . . . 6 |- ( n = k -> ( 1 / ( G ` n ) ) = ( 1 / ( G ` k ) ) )
9		simpr 110	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> k e. ( ZZ>= ` M ) )
10	2, 4	recclapd 8808	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` k ) ) e. CC )
11	6, 8, 9, 10	fvmptd3 5655	. . . . 5 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) = ( 1 / ( G ` k ) ) )
12	3, 11	sylan2 286	. . . 4 |- ( ( ph /\ k e. ( M ... N ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) = ( 1 / ( G ` k ) ) )
13	11, 10	eqeltrd 2273	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
14	1, 2, 5, 12, 13	prodfrecap 11711	. . 3 |- ( ph -> ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) = ( 1 / ( seq M ( x. , G ) ` N ) ) )
15	14	oveq2d 5938	. 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 2257	. . . . . . . . 9 |- ( k = n -> ( k e. ( ZZ>= ` M ) <-> n e. ( ZZ>= ` M ) ) )
18	17	anbi2d 464	. . . . . . . 8 |- ( k = n -> ( ( ph /\ k e. ( ZZ>= ` M ) ) <-> ( ph /\ n e. ( ZZ>= ` M ) ) ) )
19		fveq2 5558	. . . . . . . . 9 |- ( k = n -> ( G ` k ) = ( G ` n ) )
20	19	eleq1d 2265	. . . . . . . 8 |- ( k = n -> ( ( G ` k ) e. CC <-> ( G ` n ) e. CC ) )
21	18, 20	imbi12d 234	. . . . . . 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 1923	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) e. CC )
23	19	breq1d 4043	. . . . . . . 8 |- ( k = n -> ( ( G ` k ) # 0 <-> ( G ` n ) # 0 ) )
24	18, 23	imbi12d 234	. . . . . . 7 |- ( k = n -> ( ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( G ` k ) # 0 ) <-> ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 ) ) )
25	24, 4	chvarvv 1923	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 )
26	22, 25	recclapd 8808	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` n ) ) e. CC )
27	26	fmpttd 5717	. . . 4 |- ( ph -> ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) : ( ZZ>= ` M ) --> CC )
28	27	ffvelcdmda 5697	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
29	16, 2, 4	divrecapd 8820	. . . 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 5938	. . . 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 2239	. . 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 11706	. 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 2196	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
35		eluzel2 9606	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
37	34, 36, 16	prodf 11703	. . . 4 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
38	37, 1	ffvelcdmd 5698	. . 3 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
39	34, 36, 2	prodf 11703	. . . 4 |- ( ph -> seq M ( x. , G ) : ( ZZ>= ` M ) --> CC )
40	39, 1	ffvelcdmd 5698	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) e. CC )
41	1, 2, 5	prodfap0 11710	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) # 0 )
42	38, 40, 41	divrecapd 8820	. 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 2239	1 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   class class class wbr 4033   |-> cmpt 4094   ` cfv 5258  (class class class)co 5922   CCcc 7877   0cc0 7879   1c1 7880   x. cmul 7884   # cap 8608   / cdiv 8699   ZZcz 9326   ZZ>=cuz 9601   ...cfz 10083   seqcseq 10539

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-fzo 10218  df-seqfrec 10540

This theorem is referenced by: (None)