Theorem fproddivapf 11884

Description: The quotient of two finite products. A version of fproddivap 11883 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 2347	. . . 4 |- F/_ j ( B / C )
2		nfcsb1v 3125	. . . . 5 |- F/_ k [_ j / k ]_ B
3		nfcv 2347	. . . . 5 |- F/_ k /
4		nfcsb1v 3125	. . . . 5 |- F/_ k [_ j / k ]_ C
5	2, 3, 4	nfov 5973	. . . 4 |- F/_ k ( [_ j / k ]_ B / [_ j / k ]_ C )
6		csbeq1a 3101	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
7		csbeq1a 3101	. . . . 5 |- ( k = j -> C = [_ j / k ]_ C )
8	6, 7	oveq12d 5961	. . . 4 |- ( k = j -> ( B / C ) = ( [_ j / k ]_ B / [_ j / k ]_ C ) )
9	1, 5, 8	cbvprodi 11813	. . 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 1551	. . . . . 6 |- ( ph -> F/ k j e. A )
14	12, 13	nfan1 1586	. . . . 5 |- F/ k ( ph /\ j e. A )
15	2	nfel1 2358	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
16	14, 15	nfim 1594	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
17		eleq1w 2265	. . . . . 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 2273	. . . . 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 1722	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
23	4	nfel1 2358	. . . . 5 |- F/ k [_ j / k ]_ C e. CC
24	14, 23	nfim 1594	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
25	7	eleq1d 2273	. . . . 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 1722	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
29		nfcv 2347	. . . . . 6 |- F/_ k #
30		nfcv 2347	. . . . . 6 |- F/_ k 0
31	4, 29, 30	nfbr 4089	. . . . 5 |- F/ k [_ j / k ]_ C # 0
32	14, 31	nfim 1594	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
33	7	breq1d 4053	. . . . 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 1722	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
37	11, 22, 28, 36	fproddivap 11883	. 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 2347	. . . . . 6 |- F/_ j B
39	38, 2, 6	cbvprodi 11813	. . . . 5 |- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B
40	39	eqcomi 2208	. . . 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 2347	. . . . 5 |- F/_ j C
43	7	equcoms 1730	. . . . . 6 |- ( j = k -> C = [_ j / k ]_ C )
44	43	eqcomd 2210	. . . . 5 |- ( j = k -> [_ j / k ]_ C = C )
45	4, 42, 44	cbvprodi 11813	. . . 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 5961	. 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 2241	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 1372   F/wnf 1482   e. wcel 2175   [_csb 3092   class class class wbr 4043  (class class class)co 5943   Fincfn 6826   CCcc 7922   0cc0 7924   # cap 8653   / cdiv 8744   prod_cprod 11803

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-frec 6476  df-1o 6501  df-oadd 6505  df-er 6619  df-en 6827  df-dom 6828  df-fin 6829  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-fz 10130  df-fzo 10264  df-seqfrec 10591  df-exp 10682  df-ihash 10919  df-cj 11095  df-re 11096  df-im 11097  df-rsqrt 11251  df-abs 11252  df-clim 11532  df-proddc 11804

This theorem is referenced by: (None)