Theorem fproddivapf 12210

Description: The quotient of two finite products. A version of fproddivap 12209 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 2374	. . . 4 |- F/_ j ( B / C )
2		nfcsb1v 3160	. . . . 5 |- F/_ k [_ j / k ]_ B
3		nfcv 2374	. . . . 5 |- F/_ k /
4		nfcsb1v 3160	. . . . 5 |- F/_ k [_ j / k ]_ C
5	2, 3, 4	nfov 6048	. . . 4 |- F/_ k ( [_ j / k ]_ B / [_ j / k ]_ C )
6		csbeq1a 3136	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
7		csbeq1a 3136	. . . . 5 |- ( k = j -> C = [_ j / k ]_ C )
8	6, 7	oveq12d 6036	. . . 4 |- ( k = j -> ( B / C ) = ( [_ j / k ]_ B / [_ j / k ]_ C ) )
9	1, 5, 8	cbvprodi 12139	. . 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 1577	. . . . . 6 |- ( ph -> F/ k j e. A )
14	12, 13	nfan1 1612	. . . . 5 |- F/ k ( ph /\ j e. A )
15	2	nfel1 2385	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
16	14, 15	nfim 1620	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
17		eleq1w 2292	. . . . . 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 2300	. . . . 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 1748	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
23	4	nfel1 2385	. . . . 5 |- F/ k [_ j / k ]_ C e. CC
24	14, 23	nfim 1620	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
25	7	eleq1d 2300	. . . . 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 1748	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
29		nfcv 2374	. . . . . 6 |- F/_ k #
30		nfcv 2374	. . . . . 6 |- F/_ k 0
31	4, 29, 30	nfbr 4135	. . . . 5 |- F/ k [_ j / k ]_ C # 0
32	14, 31	nfim 1620	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
33	7	breq1d 4098	. . . . 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 1748	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
37	11, 22, 28, 36	fproddivap 12209	. 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 2374	. . . . . 6 |- F/_ j B
39	38, 2, 6	cbvprodi 12139	. . . . 5 |- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B
40	39	eqcomi 2235	. . . 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 2374	. . . . 5 |- F/_ j C
43	7	equcoms 1756	. . . . . 6 |- ( j = k -> C = [_ j / k ]_ C )
44	43	eqcomd 2237	. . . . 5 |- ( j = k -> [_ j / k ]_ C = C )
45	4, 42, 44	cbvprodi 12139	. . . 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 6036	. 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 2268	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 1397   F/wnf 1508   e. wcel 2202   [_csb 3127   class class class wbr 4088  (class class class)co 6018   Fincfn 6909   CCcc 8030   0cc0 8032   # cap 8761   / cdiv 8852   prod_cprod 12129

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-frec 6557  df-1o 6582  df-oadd 6586  df-er 6702  df-en 6910  df-dom 6911  df-fin 6912  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-fz 10244  df-fzo 10378  df-seqfrec 10711  df-exp 10802  df-ihash 11039  df-cj 11420  df-re 11421  df-im 11422  df-rsqrt 11576  df-abs 11577  df-clim 11857  df-proddc 12130

This theorem is referenced by: (None)