Theorem prodfdivap 12258

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 10374	. . . . 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 2234	. . . . . 6 |- ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) = ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) )
7		fveq2 5675	. . . . . . 7 |- ( n = k -> ( G ` n ) = ( G ` k ) )
8	7	oveq2d 6074	. . . . . 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 9072	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` k ) ) e. CC )
11	6, 8, 9, 10	fvmptd3 5776	. . . . 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 2311	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
14	1, 2, 5, 12, 13	prodfrecap 12257	. . 3 |- ( ph -> ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) = ( 1 / ( seq M ( x. , G ) ` N ) ) )
15	14	oveq2d 6074	. 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 2295	. . . . . . . . 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 5675	. . . . . . . . 9 |- ( k = n -> ( G ` k ) = ( G ` n ) )
20	19	eleq1d 2303	. . . . . . . 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 1960	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) e. CC )
23	19	breq1d 4124	. . . . . . . 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 1960	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 )
26	22, 25	recclapd 9072	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` n ) ) e. CC )
27	26	fmpttd 5837	. . . 4 |- ( ph -> ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) : ( ZZ>= ` M ) --> CC )
28	27	ffvelcdmda 5817	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
29	16, 2, 4	divrecapd 9084	. . . 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 6074	. . . 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 2277	. . 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 12252	. 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 2234	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
35		eluzel2 9876	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
37	34, 36, 16	prodf 12249	. . . 4 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
38	37, 1	ffvelcdmd 5818	. . 3 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
39	34, 36, 2	prodf 12249	. . . 4 |- ( ph -> seq M ( x. , G ) : ( ZZ>= ` M ) --> CC )
40	39, 1	ffvelcdmd 5818	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) e. CC )
41	1, 2, 5	prodfap0 12256	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) # 0 )
42	38, 40, 41	divrecapd 9084	. 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 2277	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 2205   class class class wbr 4114   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   CCcc 8141   0cc0 8143   1c1 8144   x. cmul 8148   # cap 8872   / cdiv 8963   ZZcz 9594   ZZ>=cuz 9871   ...cfz 10361   seqcseq 10833

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-fzo 10499  df-seqfrec 10834

This theorem is referenced by: (None)