Theorem fproddivapf 11674

Description: The quotient of two finite products. A version of fproddivap 11673 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 2332	. . . 4 |- F/_ j ( B / C )
2		nfcsb1v 3105	. . . . 5 |- F/_ k [_ j / k ]_ B
3		nfcv 2332	. . . . 5 |- F/_ k /
4		nfcsb1v 3105	. . . . 5 |- F/_ k [_ j / k ]_ C
5	2, 3, 4	nfov 5927	. . . 4 |- F/_ k ( [_ j / k ]_ B / [_ j / k ]_ C )
6		csbeq1a 3081	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
7		csbeq1a 3081	. . . . 5 |- ( k = j -> C = [_ j / k ]_ C )
8	6, 7	oveq12d 5915	. . . 4 |- ( k = j -> ( B / C ) = ( [_ j / k ]_ B / [_ j / k ]_ C ) )
9	1, 5, 8	cbvprodi 11603	. . 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 1540	. . . . . 6 |- ( ph -> F/ k j e. A )
14	12, 13	nfan1 1575	. . . . 5 |- F/ k ( ph /\ j e. A )
15	2	nfel1 2343	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
16	14, 15	nfim 1583	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
17		eleq1w 2250	. . . . . 6 |- ( k = j -> ( k e. A <-> j e. A ) )
18	17	anbi2d 464	. . . . 5 |- ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
19	6	eleq1d 2258	. . . . 5 |- ( k = j -> ( B e. CC <-> [_ j / k ]_ B e. CC ) )
20	18, 19	imbi12d 234	. . . 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 1711	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
23	4	nfel1 2343	. . . . 5 |- F/ k [_ j / k ]_ C e. CC
24	14, 23	nfim 1583	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
25	7	eleq1d 2258	. . . . 5 |- ( k = j -> ( C e. CC <-> [_ j / k ]_ C e. CC ) )
26	18, 25	imbi12d 234	. . . 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 1711	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
29		nfcv 2332	. . . . . 6 |- F/_ k #
30		nfcv 2332	. . . . . 6 |- F/_ k 0
31	4, 29, 30	nfbr 4064	. . . . 5 |- F/ k [_ j / k ]_ C # 0
32	14, 31	nfim 1583	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
33	7	breq1d 4028	. . . . 5 |- ( k = j -> ( C # 0 <-> [_ j / k ]_ C # 0 ) )
34	18, 33	imbi12d 234	. . . 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 1711	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
37	11, 22, 28, 36	fproddivap 11673	. 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 2332	. . . . . 6 |- F/_ j B
39	38, 2, 6	cbvprodi 11603	. . . . 5 |- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B
40	39	eqcomi 2193	. . . 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 2332	. . . . 5 |- F/_ j C
43	7	equcoms 1719	. . . . . 6 |- ( j = k -> C = [_ j / k ]_ C )
44	43	eqcomd 2195	. . . . 5 |- ( j = k -> [_ j / k ]_ C = C )
45	4, 42, 44	cbvprodi 11603	. . . 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 5915	. 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 2226	1 |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   F/wnf 1471   e. wcel 2160   [_csb 3072   class class class wbr 4018  (class class class)co 5897   Fincfn 6767   CCcc 7840   0cc0 7842   # cap 8569   / cdiv 8660   prod_cprod 11593

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-mulrcl 7941  ax-addcom 7942  ax-mulcom 7943  ax-addass 7944  ax-mulass 7945  ax-distr 7946  ax-i2m1 7947  ax-0lt1 7948  ax-1rid 7949  ax-0id 7950  ax-rnegex 7951  ax-precex 7952  ax-cnre 7953  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-apti 7957  ax-pre-ltadd 7958  ax-pre-mulgt0 7959  ax-pre-mulext 7960  ax-arch 7961  ax-caucvg 7962

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-ilim 4387  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-isom 5244  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-frec 6417  df-1o 6442  df-oadd 6446  df-er 6560  df-en 6768  df-dom 6769  df-fin 6770  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-sub 8161  df-neg 8162  df-reap 8563  df-ap 8570  df-div 8661  df-inn 8951  df-2 9009  df-3 9010  df-4 9011  df-n0 9208  df-z 9285  df-uz 9560  df-q 9652  df-rp 9686  df-fz 10041  df-fzo 10175  df-seqfrec 10479  df-exp 10554  df-ihash 10791  df-cj 10886  df-re 10887  df-im 10888  df-rsqrt 11042  df-abs 11043  df-clim 11322  df-proddc 11594

This theorem is referenced by: (None)