Theorem fproddivapf 12017

Description: The quotient of two finite products. A version of fproddivap 12016 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 2349	. . . 4 |- F/_ j ( B / C )
2		nfcsb1v 3130	. . . . 5 |- F/_ k [_ j / k ]_ B
3		nfcv 2349	. . . . 5 |- F/_ k /
4		nfcsb1v 3130	. . . . 5 |- F/_ k [_ j / k ]_ C
5	2, 3, 4	nfov 5987	. . . 4 |- F/_ k ( [_ j / k ]_ B / [_ j / k ]_ C )
6		csbeq1a 3106	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
7		csbeq1a 3106	. . . . 5 |- ( k = j -> C = [_ j / k ]_ C )
8	6, 7	oveq12d 5975	. . . 4 |- ( k = j -> ( B / C ) = ( [_ j / k ]_ B / [_ j / k ]_ C ) )
9	1, 5, 8	cbvprodi 11946	. . 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 1553	. . . . . 6 |- ( ph -> F/ k j e. A )
14	12, 13	nfan1 1588	. . . . 5 |- F/ k ( ph /\ j e. A )
15	2	nfel1 2360	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
16	14, 15	nfim 1596	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
17		eleq1w 2267	. . . . . 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 2275	. . . . 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 1724	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
23	4	nfel1 2360	. . . . 5 |- F/ k [_ j / k ]_ C e. CC
24	14, 23	nfim 1596	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
25	7	eleq1d 2275	. . . . 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 1724	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C e. CC )
29		nfcv 2349	. . . . . 6 |- F/_ k #
30		nfcv 2349	. . . . . 6 |- F/_ k 0
31	4, 29, 30	nfbr 4098	. . . . 5 |- F/ k [_ j / k ]_ C # 0
32	14, 31	nfim 1596	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
33	7	breq1d 4061	. . . . 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 1724	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ C # 0 )
37	11, 22, 28, 36	fproddivap 12016	. 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 2349	. . . . . 6 |- F/_ j B
39	38, 2, 6	cbvprodi 11946	. . . . 5 |- prod_ k e. A B = prod_ j e. A [_ j / k ]_ B
40	39	eqcomi 2210	. . . 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 2349	. . . . 5 |- F/_ j C
43	7	equcoms 1732	. . . . . 6 |- ( j = k -> C = [_ j / k ]_ C )
44	43	eqcomd 2212	. . . . 5 |- ( j = k -> [_ j / k ]_ C = C )
45	4, 42, 44	cbvprodi 11946	. . . 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 5975	. 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 2243	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 1373   F/wnf 1484   e. wcel 2177   [_csb 3097   class class class wbr 4051  (class class class)co 5957   Fincfn 6840   CCcc 7943   0cc0 7945   # cap 8674   / cdiv 8765   prod_cprod 11936

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-mulrcl 8044  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-0lt1 8051  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-precex 8055  ax-cnre 8056  ax-pre-ltirr 8057  ax-pre-ltwlin 8058  ax-pre-lttrn 8059  ax-pre-apti 8060  ax-pre-ltadd 8061  ax-pre-mulgt0 8062  ax-pre-mulext 8063  ax-arch 8064  ax-caucvg 8065

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-po 4351  df-iso 4352  df-iord 4421  df-on 4423  df-ilim 4424  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-isom 5289  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-frec 6490  df-1o 6515  df-oadd 6519  df-er 6633  df-en 6841  df-dom 6842  df-fin 6843  df-pnf 8129  df-mnf 8130  df-xr 8131  df-ltxr 8132  df-le 8133  df-sub 8265  df-neg 8266  df-reap 8668  df-ap 8675  df-div 8766  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-n0 9316  df-z 9393  df-uz 9669  df-q 9761  df-rp 9796  df-fz 10151  df-fzo 10285  df-seqfrec 10615  df-exp 10706  df-ihash 10943  df-cj 11228  df-re 11229  df-im 11230  df-rsqrt 11384  df-abs 11385  df-clim 11665  df-proddc 11937

This theorem is referenced by: (None)