Theorem fsummulc1f 39802

Description: Closure of a finite sum of complex numbers A ( k ). A version of fsummulc1 14517 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

Hypotheses

Ref	Expression
fsummulc1f.ph	|- F/ k ph
fsummulclf.a	|- ( ph -> A e. Fin )
fsummulclf.c	|- ( ph -> C e. CC )
fsummulclf.b	|- ( ( ph /\ k e. A ) -> B e. CC )

Assertion

Ref	Expression
fsummulc1f	|- ( ph -> ( sum_ k e. A B x. C ) = sum_ k e. A ( B x. C ) )

Distinct variable groups:   A, k   C, k

Allowed substitution hints:   ph( k)   B( k)

Proof of Theorem fsummulc1f

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1a 3542	. . . . 5 |- ( k = j -> B = [_ j / k ]_ B )
2		nfcv 2764	. . . . 5 |- F/_ j A
3		nfcv 2764	. . . . 5 |- F/_ k A
4		nfcv 2764	. . . . 5 |- F/_ j B
5		nfcsb1v 3549	. . . . 5 |- F/_ k [_ j / k ]_ B
6	1, 2, 3, 4, 5	cbvsum 14425	. . . 4 |- sum_ k e. A B = sum_ j e. A [_ j / k ]_ B
7	6	oveq1i 6660	. . 3 |- ( sum_ k e. A B x. C ) = ( sum_ j e. A [_ j / k ]_ B x. C )
8	7	a1i 11	. 2 |- ( ph -> ( sum_ k e. A B x. C ) = ( sum_ j e. A [_ j / k ]_ B x. C ) )
9		fsummulclf.a	. . 3 |- ( ph -> A e. Fin )
10		fsummulclf.c	. . 3 |- ( ph -> C e. CC )
11		fsummulc1f.ph	. . . . . 6 |- F/ k ph
12		nfv 1843	. . . . . 6 |- F/ k j e. A
13	11, 12	nfan 1828	. . . . 5 |- F/ k ( ph /\ j e. A )
14	5	nfel1 2779	. . . . 5 |- F/ k [_ j / k ]_ B e. CC
15	13, 14	nfim 1825	. . . 4 |- F/ k ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
16		eleq1 2689	. . . . . 6 |- ( k = j -> ( k e. A <-> j e. A ) )
17	16	anbi2d 740	. . . . 5 |- ( k = j -> ( ( ph /\ k e. A ) <-> ( ph /\ j e. A ) ) )
18	1	eleq1d 2686	. . . . 5 |- ( k = j -> ( B e. CC <-> [_ j / k ]_ B e. CC ) )
19	17, 18	imbi12d 334	. . . 4 |- ( k = j -> ( ( ( ph /\ k e. A ) -> B e. CC ) <-> ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC ) ) )
20		fsummulclf.b	. . . 4 |- ( ( ph /\ k e. A ) -> B e. CC )
21	15, 19, 20	chvar 2262	. . 3 |- ( ( ph /\ j e. A ) -> [_ j / k ]_ B e. CC )
22	9, 10, 21	fsummulc1 14517	. 2 |- ( ph -> ( sum_ j e. A [_ j / k ]_ B x. C ) = sum_ j e. A ( [_ j / k ]_ B x. C ) )
23		eqcom 2629	. . . . . . . 8 |- ( k = j <-> j = k )
24	23	imbi1i 339	. . . . . . 7 |- ( ( k = j -> B = [_ j / k ]_ B ) <-> ( j = k -> B = [_ j / k ]_ B ) )
25		eqcom 2629	. . . . . . . 8 |- ( B = [_ j / k ]_ B <-> [_ j / k ]_ B = B )
26	25	imbi2i 326	. . . . . . 7 |- ( ( j = k -> B = [_ j / k ]_ B ) <-> ( j = k -> [_ j / k ]_ B = B ) )
27	24, 26	bitri 264	. . . . . 6 |- ( ( k = j -> B = [_ j / k ]_ B ) <-> ( j = k -> [_ j / k ]_ B = B ) )
28	1, 27	mpbi 220	. . . . 5 |- ( j = k -> [_ j / k ]_ B = B )
29	28	oveq1d 6665	. . . 4 |- ( j = k -> ( [_ j / k ]_ B x. C ) = ( B x. C ) )
30		nfcv 2764	. . . . 5 |- F/_ k x.
31		nfcv 2764	. . . . 5 |- F/_ k C
32	5, 30, 31	nfov 6676	. . . 4 |- F/_ k ( [_ j / k ]_ B x. C )
33		nfcv 2764	. . . 4 |- F/_ j ( B x. C )
34	29, 3, 2, 32, 33	cbvsum 14425	. . 3 |- sum_ j e. A ( [_ j / k ]_ B x. C ) = sum_ k e. A ( B x. C )
35	34	a1i 11	. 2 |- ( ph -> sum_ j e. A ( [_ j / k ]_ B x. C ) = sum_ k e. A ( B x. C ) )
36	8, 22, 35	3eqtrd 2660	1 |- ( ph -> ( sum_ k e. A B x. C ) = sum_ k e. A ( B x. C ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   F/wnf 1708   e. wcel 1990   [_csb 3533  (class class class)co 6650   Fincfn 7955   CCcc 9934   x. cmul 9941   sum_csu 14416

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417

This theorem is referenced by:  dvmptfprodlem  40159