Theorem fproddivapf 11566

Description: The quotient of two finite products. A version of fproddivap 11565 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
fproddivf.kph	|- F/ k ph
fproddivf.a	|- ( ph -> A e. Fin )
fproddivf.b	|- ( ( ph /\ k e. A ) -> B e. CC )
fproddivf.c	|- ( ( ph /\ k e. A ) -> C e. CC )
fproddivf.ap0	|- ( ( ph /\ k e. A ) -> C # 0 )

Assertion

Ref	Expression
fproddivapf	|- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )

Distinct variable group:   A, k

Allowed substitution hints:   ph( k)   B( k)   C( k)

Proof of Theorem fproddivapf

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2306	. . . 4 |- F/_ j ( B / C )
2		nfcsb1v 3076	. . . . 5 |- F/_ k [_ j / k ]_ B
3		nfcv 2306	. . . . 5 |- F/_ k /
4		nfcsb1v 3076	. . . . 5 |- F/_ k [_ j / k ]_ C
5	2, 3, 4	nfov 5866	. . . 4 |- F/_ k ( [_ j / k ]_ B / [_ j / k ]_ C )
6		csbeq1a 3052	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
7		csbeq1a 3052	. . . . 5 |- ( k = j -> C = [_ j / k ]_ C )
8	6, 7	oveq12d 5857	. . . 4 |- ( k = j -> ( B / C ) = ( [_ j / k ]_ B / [_ j / k ]_ C ) )
9	1, 5, 8	cbvprodi 11495	. . 3 |- prod_ k e. A ( B / C ) = prod_ j e. A ( [_ j / k ]_ B / [_ j / k ]_ C )
10	9	a1i 9	. 2 |- ( ph -> prod_ k e. A ( B / C ) = prod_ j e. A ( [_ j / k ]_ B / [_ j / k ]_ C ) )
11		fproddivf.a	. . 3 |- ( ph -> A e. Fin )
12		fproddivf.kph	. . . . . 6 |- F/ k ph
13		nfvd 1516	. . . . . 6 |- ( ph -> F/ k j e. A )
14	12, 13	nfan1 1551	. . . . 5 |- F/ k ( ph /\ j e. A )
15	2	nfel1 2317	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
16	14, 15	nfim 1559	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
17		eleq1w 2225	. . . . . 6 |- ( k = j -> ( k e. A <-> j e. A ) )
18	17	anbi2d 460	. . . . 5 |- ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
19	6	eleq1d 2233	. . . . 5 |- ( k = j -> ( B e. CC <-> [_ j / k ]_ B e. CC ) )
20	18, 19	imbi12d 233	. . . 4 |- ( k = j -> ( ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC ) ) )
21		fproddivf.b	. . . 4 |- ( ( ph /\ k e. A ) -> B e. CC )
22	16, 20, 21	chvarfv 1687	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
23	4	nfel1 2317	. . . . 5 |- F/ k [_ j / k ]_ C e. CC
24	14, 23	nfim 1559	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
25	7	eleq1d 2233	. . . . 5 |- ( k = j -> ( C e. CC <-> [_ j / k ]_ C e. CC ) )
26	18, 25	imbi12d 233	. . . 4 |- ( k = j -> ( ( ( ph /\ k e. A ) -> C e. CC ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC ) ) )
27		fproddivf.c	. . . 4 |- ( ( ph /\ k e. A ) -> C e. CC )
28	24, 26, 27	chvarfv 1687	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
29		nfcv 2306	. . . . . 6 |- F/_ k #
30		nfcv 2306	. . . . . 6 |- F/_ k 0
31	4, 29, 30	nfbr 4025	. . . . 5 |- F/ k [_ j / k ]_ C # 0
32	14, 31	nfim 1559	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
33	7	breq1d 3989	. . . . 5 |- ( k = j -> ( C # 0 <-> [_ j / k ]_ C # 0 ) )
34	18, 33	imbi12d 233	. . . 4 |- ( k = j -> ( ( ( ph /\ k e. A ) -> C # 0 ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 ) ) )
35		fproddivf.ap0	. . . 4 |- ( ( ph /\ k e. A ) -> C # 0 )
36	32, 34, 35	chvarfv 1687	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
37	11, 22, 28, 36	fproddivap 11565	. 2 |- ( ph -> prod_ j e. A ( [_ j / k ]_ B / [_ j / k ]_ C ) = ( prod_ j e. A [_ j / k ]_ B / prod_ j e. A [_ j / k ]_ C ) )
38		nfcv 2306	. . . . . 6 |- F/_ j B
39	38, 2, 6	cbvprodi 11495	. . . . 5 |- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B
40	39	eqcomi 2168	. . . 4 |- prod_ j e. A [_ j / k ]_ B = prod_ k e. A B
41	40	a1i 9	. . 3 |- ( ph -> prod_ j e. A [_ j / k ]_ B = prod_ k e. A B )
42		nfcv 2306	. . . . 5 |- F/_ j C
43	7	equcoms 1695	. . . . . 6 |- ( j = k -> C = [_ j / k ]_ C )
44	43	eqcomd 2170	. . . . 5 |- ( j = k -> [_ j / k ]_ C = C )
45	4, 42, 44	cbvprodi 11495	. . . 4 |- prod_ j e. A [_ j / k ]_ C = prod_ k e. A C
46	45	a1i 9	. . 3 |- ( ph -> prod_ j e. A [_ j / k ]_ C = prod_ k e. A C )
47	41, 46	oveq12d 5857	. 2 |- ( ph -> ( prod_ j e. A [_ j / k ]_ B / prod_ j e. A [_ j / k ]_ C ) = ( prod_ k e. A B / prod_ k e. A C ) )
48	10, 37, 47	3eqtrd 2201	1 |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1342   F/wnf 1447   e. wcel 2135   [_csb 3043   class class class wbr 3979  (class class class)co 5839   Fincfn 6700   CCcc 7745   0cc0 7747   # cap 8473   / cdiv 8562   prod_cprod 11485

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4094  ax-sep 4097  ax-nul 4105  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-iinf 4562  ax-cnex 7838  ax-resscn 7839  ax-1cn 7840  ax-1re 7841  ax-icn 7842  ax-addcl 7843  ax-addrcl 7844  ax-mulcl 7845  ax-mulrcl 7846  ax-addcom 7847  ax-mulcom 7848  ax-addass 7849  ax-mulass 7850  ax-distr 7851  ax-i2m1 7852  ax-0lt1 7853  ax-1rid 7854  ax-0id 7855  ax-rnegex 7856  ax-precex 7857  ax-cnre 7858  ax-pre-ltirr 7859  ax-pre-ltwlin 7860  ax-pre-lttrn 7861  ax-pre-apti 7862  ax-pre-ltadd 7863  ax-pre-mulgt0 7864  ax-pre-mulext 7865  ax-arch 7866  ax-caucvg 7867

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rmo 2450  df-rab 2451  df-v 2726  df-sbc 2950  df-csb 3044  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3408  df-if 3519  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-int 3822  df-iun 3865  df-br 3980  df-opab 4041  df-mpt 4042  df-tr 4078  df-id 4268  df-po 4271  df-iso 4272  df-iord 4341  df-on 4343  df-ilim 4344  df-suc 4346  df-iom 4565  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-rn 4612  df-res 4613  df-ima 4614  df-iota 5150  df-fun 5187  df-fn 5188  df-f 5189  df-f1 5190  df-fo 5191  df-f1o 5192  df-fv 5193  df-isom 5194  df-riota 5795  df-ov 5842  df-oprab 5843  df-mpo 5844  df-1st 6103  df-2nd 6104  df-recs 6267  df-irdg 6332  df-frec 6353  df-1o 6378  df-oadd 6382  df-er 6495  df-en 6701  df-dom 6702  df-fin 6703  df-pnf 7929  df-mnf 7930  df-xr 7931  df-ltxr 7932  df-le 7933  df-sub 8065  df-neg 8066  df-reap 8467  df-ap 8474  df-div 8563  df-inn 8852  df-2 8910  df-3 8911  df-4 8912  df-n0 9109  df-z 9186  df-uz 9461  df-q 9552  df-rp 9584  df-fz 9939  df-fzo 10072  df-seqfrec 10375  df-exp 10449  df-ihash 10683  df-cj 10778  df-re 10779  df-im 10780  df-rsqrt 10934  df-abs 10935  df-clim 11214  df-proddc 11486

This theorem is referenced by: (None)