Theorem iprodmul 14734

Description: Multiplication of infinite sums. (Contributed by Scott Fenton, 18-Dec-2017.)

Hypotheses

Ref	Expression
iprodmul.1	|- Z = ( ZZ>= ` M )
iprodmul.2	|- ( ph -> M e. ZZ )
iprodmul.3	|- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) )
iprodmul.4	|- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
iprodmul.5	|- ( ( ph /\ k e. Z ) -> A e. CC )
iprodmul.6	|- ( ph -> E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) )
iprodmul.7	|- ( ( ph /\ k e. Z ) -> ( G ` k ) = B )
iprodmul.8	|- ( ( ph /\ k e. Z ) -> B e. CC )

Assertion

Ref	Expression
iprodmul	|- ( ph -> prod_ k e. Z ( A x. B ) = ( prod_ k e. Z A x. prod_ k e. Z B ) )

Distinct variable groups:   A, n, y   B, m, z   k, F, m, n, y, z   k, G, m, n, y, z   ph, k, y, z   k, M, m, n   ph, m, y   y, M   z, m, M   ph, n, y, z   k, Z, m, n, y, z

Allowed substitution hints:   A( z, k, m)   B( y, k, n)

Proof of Theorem iprodmul

Dummy variables j a p w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iprodmul.1	. 2 |- Z = ( ZZ>= ` M )
2		iprodmul.2	. 2 |- ( ph -> M e. ZZ )
3		iprodmul.3	. . . 4 |- ( ph -> E. n e. Z E. y ( y =/= 0 /\ seq n ( x. , F ) ~~> y ) )
4		iprodmul.4	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( F ` k ) = A )
5		iprodmul.5	. . . . 5 |- ( ( ph /\ k e. Z ) -> A e. CC )
6	4, 5	eqeltrd 2701	. . . 4 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. CC )
7		iprodmul.6	. . . 4 |- ( ph -> E. m e. Z E. z ( z =/= 0 /\ seq m ( x. , G ) ~~> z ) )
8		iprodmul.7	. . . . 5 |- ( ( ph /\ k e. Z ) -> ( G ` k ) = B )
9		iprodmul.8	. . . . 5 |- ( ( ph /\ k e. Z ) -> B e. CC )
10	8, 9	eqeltrd 2701	. . . 4 |- ( ( ph /\ k e. Z ) -> ( G ` k ) e. CC )
11		fveq2 6191	. . . . . . 7 |- ( a = k -> ( F ` a ) = ( F ` k ) )
12		fveq2 6191	. . . . . . 7 |- ( a = k -> ( G ` a ) = ( G ` k ) )
13	11, 12	oveq12d 6668	. . . . . 6 |- ( a = k -> ( ( F ` a ) x. ( G ` a ) ) = ( ( F ` k ) x. ( G ` k ) ) )
14		eqid 2622	. . . . . 6 |- ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) = ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) )
15		ovex 6678	. . . . . 6 |- ( ( F ` k ) x. ( G ` k ) ) e. _V
16	13, 14, 15	fvmpt 6282	. . . . 5 |- ( k e. Z -> ( ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
17	16	adantl 482	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
18	1, 3, 6, 7, 10, 17	ntrivcvgmul 14634	. . 3 |- ( ph -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ) ~~> w ) )
19		fveq2 6191	. . . . . . . . . 10 |- ( m = a -> ( F ` m ) = ( F ` a ) )
20		fveq2 6191	. . . . . . . . . 10 |- ( m = a -> ( G ` m ) = ( G ` a ) )
21	19, 20	oveq12d 6668	. . . . . . . . 9 |- ( m = a -> ( ( F ` m ) x. ( G ` m ) ) = ( ( F ` a ) x. ( G ` a ) ) )
22	21	cbvmptv 4750	. . . . . . . 8 |- ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) = ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) )
23		seqeq3 12806	. . . . . . . 8 |- ( ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) = ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) -> seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) = seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ) )
24	22, 23	ax-mp 5	. . . . . . 7 |- seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) = seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) )
25	24	breq1i 4660	. . . . . 6 |- ( seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ~~> w <-> seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ) ~~> w )
26	25	anbi2i 730	. . . . 5 |- ( ( w =/= 0 /\ seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ~~> w ) <-> ( w =/= 0 /\ seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ) ~~> w ) )
27	26	exbii 1774	. . . 4 |- ( E. w ( w =/= 0 /\ seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ~~> w ) <-> E. w ( w =/= 0 /\ seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ) ~~> w ) )
28	27	rexbii 3041	. . 3 |- ( E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ~~> w ) <-> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , ( a e. Z |-> ( ( F ` a ) x. ( G ` a ) ) ) ) ~~> w ) )
29	18, 28	sylibr 224	. 2 |- ( ph -> E. p e. Z E. w ( w =/= 0 /\ seq p ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ~~> w ) )
30		simpr 477	. . . 4 |- ( ( ph /\ k e. Z ) -> k e. Z )
31	6, 10	mulcld 10060	. . . 4 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) x. ( G ` k ) ) e. CC )
32		fveq2 6191	. . . . . 6 |- ( m = k -> ( F ` m ) = ( F ` k ) )
33		fveq2 6191	. . . . . 6 |- ( m = k -> ( G ` m ) = ( G ` k ) )
34	32, 33	oveq12d 6668	. . . . 5 |- ( m = k -> ( ( F ` m ) x. ( G ` m ) ) = ( ( F ` k ) x. ( G ` k ) ) )
35		eqid 2622	. . . . 5 |- ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) = ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) )
36	34, 35	fvmptg 6280	. . . 4 |- ( ( k e. Z /\ ( ( F ` k ) x. ( G ` k ) ) e. CC ) -> ( ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
37	30, 31, 36	syl2anc 693	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
38	4, 8	oveq12d 6668	. . 3 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) x. ( G ` k ) ) = ( A x. B ) )
39	37, 38	eqtrd 2656	. 2 |- ( ( ph /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ` k ) = ( A x. B ) )
40	5, 9	mulcld 10060	. 2 |- ( ( ph /\ k e. Z ) -> ( A x. B ) e. CC )
41	1, 2, 3, 4, 5	iprodclim2 14730	. . 3 |- ( ph -> seq M ( x. , F ) ~~> prod_ k e. Z A )
42		seqex 12803	. . . 4 |- seq M ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) e. _V
43	42	a1i 11	. . 3 |- ( ph -> seq M ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) e. _V )
44	1, 2, 7, 8, 9	iprodclim2 14730	. . 3 |- ( ph -> seq M ( x. , G ) ~~> prod_ k e. Z B )
45	1, 2, 6	prodf 14619	. . . 4 |- ( ph -> seq M ( x. , F ) : Z --> CC )
46	45	ffvelrnda 6359	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , F ) ` j ) e. CC )
47	1, 2, 10	prodf 14619	. . . 4 |- ( ph -> seq M ( x. , G ) : Z --> CC )
48	47	ffvelrnda 6359	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , G ) ` j ) e. CC )
49		simpr 477	. . . . 5 |- ( ( ph /\ j e. Z ) -> j e. Z )
50	49, 1	syl6eleq 2711	. . . 4 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
51		elfzuz 12338	. . . . . . 7 |- ( k e. ( M ... j ) -> k e. ( ZZ>= ` M ) )
52	51, 1	syl6eleqr 2712	. . . . . 6 |- ( k e. ( M ... j ) -> k e. Z )
53	52, 6	sylan2 491	. . . . 5 |- ( ( ph /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
54	53	adantlr 751	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( F ` k ) e. CC )
55	52, 10	sylan2 491	. . . . 5 |- ( ( ph /\ k e. ( M ... j ) ) -> ( G ` k ) e. CC )
56	55	adantlr 751	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( G ` k ) e. CC )
57	37	adantlr 751	. . . . 5 |- ( ( ( ph /\ j e. Z ) /\ k e. Z ) -> ( ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
58	52, 57	sylan2 491	. . . 4 |- ( ( ( ph /\ j e. Z ) /\ k e. ( M ... j ) ) -> ( ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ` k ) = ( ( F ` k ) x. ( G ` k ) ) )
59	50, 54, 56, 58	prodfmul 14622	. . 3 |- ( ( ph /\ j e. Z ) -> ( seq M ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ` j ) = ( ( seq M ( x. , F ) ` j ) x. ( seq M ( x. , G ) ` j ) ) )
60	1, 2, 41, 43, 44, 46, 48, 59	climmul 14363	. 2 |- ( ph -> seq M ( x. , ( m e. Z |-> ( ( F ` m ) x. ( G ` m ) ) ) ) ~~> ( prod_ k e. Z A x. prod_ k e. Z B ) )
61	1, 2, 29, 39, 40, 60	iprodclim 14729	1 |- ( ph -> prod_ k e. Z ( A x. B ) = ( prod_ k e. Z A x. prod_ k e. Z B ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   x. cmul 9941   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   seqcseq 12801   ~~> cli 14215   prod_cprod 14635

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-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-prod 14636

This theorem is referenced by: (None)