Theorem iserzmulc1 7072

Description: Multiplication of an infinite series by a constant. (Contributed by Paul Chapman, 14-Nov-2007.)

Hypotheses

Ref	Expression
iserzmulc1.1	|- A e. V
iserzmulc1.2	|- F e. V
iserzmulc1.3	|- G e. V

Assertion

Ref	Expression
iserzmulc1	|- ((M e. ZZ /\ C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq F) ~~> A -> (<.M, + >. seq G) ~~> (C x. A)))

Distinct variable groups:   C,k   k,F   k,G   k,M

Proof of Theorem iserzmulc1

Step	Hyp	Ref	Expression
1		oprex 3968	. . . . . . 7 |- (<.M, + >. seq F) e. V
2		oprex 3968	. . . . . . 7 |- (<.M, + >. seq G) e. V
3		iserzmulc1.1	. . . . . . 7 |- A e. V
4	1, 2, 3	climmulc2 7065	. . . . . 6 |- (((C e. CC /\ (<.M, + >. seq F) ~~> A) /\ (M e. ZZ /\ A.m e. (ZZ>` M)(((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))))) -> (<.M, + >. seq G) ~~> (C x. A))
5	4	expcom 374	. . . . 5 |- ((M e. ZZ /\ A.m e. (ZZ>` M)(((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m)))) -> ((C e. CC /\ (<.M, + >. seq F) ~~> A) -> (<.M, + >. seq G) ~~> (C x. A)))
6	5	exp4b 379	. . . 4 |- (M e. ZZ -> (A.m e. (ZZ>` M)(((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))) -> (C e. CC -> ((<.M, + >. seq F) ~~> A -> (<.M, + >. seq G) ~~> (C x. A)))))
7	6	imp3a 361	. . 3 |- (M e. ZZ -> ((A.m e. (ZZ>` M)(((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))) /\ C e. CC) -> ((<.M, + >. seq F) ~~> A -> (<.M, + >. seq G) ~~> (C x. A))))
8		iserzmulc1.2	. . . . . . . . . 10 |- F e. V
9	8	serzcl2t 6987	. . . . . . . . 9 |- ((m e. (ZZ>` M) /\ A.k e. (ZZ>` M)(F` k) e. CC) -> ((<.M, + >. seq F)` m) e. CC)
10		pm3.26 319	. . . . . . . . . 10 |- (((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> (F` k) e. CC)
11	10	r19.20si 1698	. . . . . . . . 9 |- (A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> A.k e. (ZZ>` M)(F` k) e. CC)
12	9, 11	sylan2 451	. . . . . . . 8 |- ((m e. (ZZ>` M) /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq F)` m) e. CC)
13	12	expcom 374	. . . . . . 7 |- (A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> (m e. (ZZ>` M) -> ((<.M, + >. seq F)` m) e. CC))
14	13	adantl 388	. . . . . 6 |- ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (m e. (ZZ>` M) -> ((<.M, + >. seq F)` m) e. CC))
15		iserzmulc1.3	. . . . . . . . . . 11 |- G e. V
16	8, 15	serzmulc1 6995	. . . . . . . . . 10 |- ((m e. (ZZ>` M) /\ C e. CC /\ A.k e. (M...m)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (C x. ((<.M, + >. seq F)` m)) = ((<.M, + >. seq G)` m))
17	16	eqcomd 1472	. . . . . . . . 9 |- ((m e. (ZZ>` M) /\ C e. CC /\ A.k e. (M...m)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m)))
18		elfzuzt 6420	. . . . . . . . . . 11 |- (k e. (M...m) -> k e. (ZZ>` M))
19	18	imim1i 16	. . . . . . . . . 10 |- ((k e. (ZZ>` M) -> ((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (k e. (M...m) -> ((F` k) e. CC /\ (G` k) = (C x. (F` k)))))
20	19	r19.20i2 1695	. . . . . . . . 9 |- (A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k))) -> A.k e. (M...m)((F` k) e. CC /\ (G` k) = (C x. (F` k))))
21	17, 20	syl3an3 859	. . . . . . . 8 |- ((m e. (ZZ>` M) /\ C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m)))
22	21	3expib 834	. . . . . . 7 |- (m e. (ZZ>` M) -> ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))))
23	22	com12 11	. . . . . 6 |- ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (m e. (ZZ>` M) -> ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))))
24	14, 23	jcad 598	. . . . 5 |- ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (m e. (ZZ>` M) -> (((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m)))))
25	24	r19.21aiv 1705	. . . 4 |- ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> A.m e. (ZZ>` M)(((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))))
26		pm3.26 319	. . . 4 |- ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> C e. CC)
27	25, 26	jca 288	. . 3 |- ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> (A.m e. (ZZ>` M)(((<.M, + >. seq F)` m) e. CC /\ ((<.M, + >. seq G)` m) = (C x. ((<.M, + >. seq F)` m))) /\ C e. CC))
28	7, 27	syl5 21	. 2 |- (M e. ZZ -> ((C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq F) ~~> A -> (<.M, + >. seq G) ~~> (C x. A))))
29	28	3impib 829	1 |- ((M e. ZZ /\ C e. CC /\ A.k e. (ZZ>` M)((F` k) e. CC /\ (G` k) = (C x. (F` k)))) -> ((<.M, + >. seq F) ~~> A -> (<.M, + >. seq G) ~~> (C x. A)))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802  <.cop 2401   class class class wbr 2609  ` cfv 3172  (class class class)co 3948  CCcc 5204   + caddc 5209   x. cmul 5211  ZZcz 5270  ZZ>cuz 6349  ...cfz 6399   seq cseqz 6463   ~~> cli 6912

This theorem is referenced by:  isummulc1 7147  ef1tllem 7323  ef01tllem1 7325

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943