Theorem prodfdivap 12233

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 10355	. . . . 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 2232	. . . . . 6 |- ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) = ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) )
7		fveq2 5670	. . . . . . 7 |- ( n = k -> ( G ` n ) = ( G ` k ) )
8	7	oveq2d 6066	. . . . . 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 9055	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` k ) ) e. CC )
11	6, 8, 9, 10	fvmptd3 5771	. . . . 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 2309	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
14	1, 2, 5, 12, 13	prodfrecap 12232	. . 3 |- ( ph -> ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) = ( 1 / ( seq M ( x. , G ) ` N ) ) )
15	14	oveq2d 6066	. 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 2293	. . . . . . . . 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 5670	. . . . . . . . 9 |- ( k = n -> ( G ` k ) = ( G ` n ) )
20	19	eleq1d 2301	. . . . . . . 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 1958	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) e. CC )
23	19	breq1d 4119	. . . . . . . 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 1958	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 )
26	22, 25	recclapd 9055	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` n ) ) e. CC )
27	26	fmpttd 5832	. . . 4 |- ( ph -> ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) : ( ZZ>= ` M ) --> CC )
28	27	ffvelcdmda 5812	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
29	16, 2, 4	divrecapd 9067	. . . 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 6066	. . . 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 2275	. . 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 12227	. 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 2232	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
35		eluzel2 9858	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
37	34, 36, 16	prodf 12224	. . . 4 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
38	37, 1	ffvelcdmd 5813	. . 3 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
39	34, 36, 2	prodf 12224	. . . 4 |- ( ph -> seq M ( x. , G ) : ( ZZ>= ` M ) --> CC )
40	39, 1	ffvelcdmd 5813	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) e. CC )
41	1, 2, 5	prodfap0 12231	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) # 0 )
42	38, 40, 41	divrecapd 9067	. 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 2275	1 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   class class class wbr 4109   |-> cmpt 4171   ` cfv 5352  (class class class)co 6050   CCcc 8125   0cc0 8127   1c1 8128   x. cmul 8132   # cap 8855   / cdiv 8946   ZZcz 9577   ZZ>=cuz 9853   ...cfz 10342   seqcseq 10809

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-seqfrec 10810

This theorem is referenced by: (None)