Theorem fproddivapf 12423

Description: The quotient of two finite products. A version of fproddivap 12422 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 2392	. . . 4 |- F/_ j ( B / C )
2		nfcsb1v 3180	. . . . 5 |- F/_ k [_ j / k ]_ B
3		nfcv 2392	. . . . 5 |- F/_ k /
4		nfcsb1v 3180	. . . . 5 |- F/_ k [_ j / k ]_ C
5	2, 3, 4	nfov 6118	. . . 4 |- F/_ k ( [_ j / k ]_ B / [_ j / k ]_ C )
6		csbeq1a 3156	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
7		csbeq1a 3156	. . . . 5 |- ( k = j -> C = [_ j / k ]_ C )
8	6, 7	oveq12d 6106	. . . 4 |- ( k = j -> ( B / C ) = ( [_ j / k ]_ B / [_ j / k ]_ C ) )
9	1, 5, 8	cbvprodi 12352	. . 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 1582	. . . . . 6 |- ( ph -> F/ k j e. A )
14	12, 13	nfan1 1617	. . . . 5 |- F/ k ( ph /\ j e. A )
15	2	nfel1 2403	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
16	14, 15	nfim 1625	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
17		eleq1w 2299	. . . . . 6 |- ( k = j -> ( k e. A <-> j e. A ) )
18	17	anbi2d 468	. . . . 5 |- ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
19	6	eleq1d 2307	. . . . 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 1752	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
23	4	nfel1 2403	. . . . 5 |- F/ k [_ j / k ]_ C e. CC
24	14, 23	nfim 1625	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
25	7	eleq1d 2307	. . . . 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 1752	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
29		nfcv 2392	. . . . . 6 |- F/_ k #
30		nfcv 2392	. . . . . 6 |- F/_ k 0
31	4, 29, 30	nfbr 4178	. . . . 5 |- F/ k [_ j / k ]_ C # 0
32	14, 31	nfim 1625	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
33	7	breq1d 4141	. . . . 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 1752	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
37	11, 22, 28, 36	fproddivap 12422	. 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 2392	. . . . . 6 |- F/_ j B
39	38, 2, 6	cbvprodi 12352	. . . . 5 |- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B
40	39	eqcomi 2242	. . . 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 2392	. . . . 5 |- F/_ j C
43	7	equcoms 1760	. . . . . 6 |- ( j = k -> C = [_ j / k ]_ C )
44	43	eqcomd 2244	. . . . 5 |- ( j = k -> [_ j / k ]_ C = C )
45	4, 42, 44	cbvprodi 12352	. . . 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 6106	. 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 2275	1 |- ( ph -> prod_ k e. A ( B / C ) = ( prod_ k e. A B / prod_ k e. A C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   F/wnf 1513   e. wcel 2209   [_csb 3147   class class class wbr 4131  (class class class)co 6088   Fincfn 7026   CCcc 8183   0cc0 8185   # cap 8917   / cdiv 9010   prod_cprod 12342

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304  ax-caucvg 8305

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-isom 5389  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-uz 9938  df-q 10036  df-rp 10072  df-fz 10429  df-fzo 10567  df-seqfrec 10906  df-exp 10997  df-ihash 11237  df-cj 11629  df-re 11630  df-im 11631  df-rsqrt 11786  df-abs 11787  df-clim 12070  df-proddc 12343

This theorem is used by: (None)