Theorem prodfdivap 12073

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 10229	. . . . 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 2229	. . . . . 6 |- ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) = ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) )
7		fveq2 5629	. . . . . . 7 |- ( n = k -> ( G ` n ) = ( G ` k ) )
8	7	oveq2d 6023	. . . . . 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 8939	. . . . . 6 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` k ) ) e. CC )
11	6, 8, 9, 10	fvmptd3 5730	. . . . 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 2306	. . . 4 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
14	1, 2, 5, 12, 13	prodfrecap 12072	. . 3 |- ( ph -> ( seq M ( x. , ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ) ` N ) = ( 1 / ( seq M ( x. , G ) ` N ) ) )
15	14	oveq2d 6023	. 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 2290	. . . . . . . . 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 5629	. . . . . . . . 9 |- ( k = n -> ( G ` k ) = ( G ` n ) )
20	19	eleq1d 2298	. . . . . . . 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 1955	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) e. CC )
23	19	breq1d 4093	. . . . . . . 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 1955	. . . . . 6 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( G ` n ) # 0 )
26	22, 25	recclapd 8939	. . . . 5 |- ( ( ph /\ n e. ( ZZ>= ` M ) ) -> ( 1 / ( G ` n ) ) e. CC )
27	26	fmpttd 5792	. . . 4 |- ( ph -> ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) : ( ZZ>= ` M ) --> CC )
28	27	ffvelcdmda 5772	. . 3 |- ( ( ph /\ k e. ( ZZ>= ` M ) ) -> ( ( n e. ( ZZ>= ` M ) |-> ( 1 / ( G ` n ) ) ) ` k ) e. CC )
29	16, 2, 4	divrecapd 8951	. . . 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 6023	. . . 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 2272	. . 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 12067	. 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 2229	. . . . 5 |- ( ZZ>= ` M ) = ( ZZ>= ` M )
35		eluzel2 9738	. . . . . 6 |- ( N e. ( ZZ>= ` M ) -> M e. ZZ )
36	1, 35	syl 14	. . . . 5 |- ( ph -> M e. ZZ )
37	34, 36, 16	prodf 12064	. . . 4 |- ( ph -> seq M ( x. , F ) : ( ZZ>= ` M ) --> CC )
38	37, 1	ffvelcdmd 5773	. . 3 |- ( ph -> ( seq M ( x. , F ) ` N ) e. CC )
39	34, 36, 2	prodf 12064	. . . 4 |- ( ph -> seq M ( x. , G ) : ( ZZ>= ` M ) --> CC )
40	39, 1	ffvelcdmd 5773	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) e. CC )
41	1, 2, 5	prodfap0 12071	. . 3 |- ( ph -> ( seq M ( x. , G ) ` N ) # 0 )
42	38, 40, 41	divrecapd 8951	. 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 2272	1 |- ( ph -> ( seq M ( x. , H ) ` N ) = ( ( seq M ( x. , F ) ` N ) / ( seq M ( x. , G ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   class class class wbr 4083   |-> cmpt 4145   ` cfv 5318  (class class class)co 6007   CCcc 8008   0cc0 8010   1c1 8011   x. cmul 8015   # cap 8739   / cdiv 8830   ZZcz 9457   ZZ>=cuz 9733   ...cfz 10216   seqcseq 10681

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-seqfrec 10682

This theorem is referenced by: (None)