Theorem iserzmulc1 7136

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 3989	. . . . . . 7 |- (<.M, + >. seq F) e. V
2		oprex 3989	. . . . . . 7 |- (<.M, + >. seq G) e. V
3		iserzmulc1.1	. . . . . . 7 |- A e. V
4	1, 2, 3	climmulc2 7129	. . . . . 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 381	. . . 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 7049	. . . . . . . . 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 1709	. . . . . . . . 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 453	. . . . . . . 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 390	. . . . . 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 7057	. . . . . . . . . 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 1483	. . . . . . . . 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 6489	. . . . . . . . . . 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 1706	. . . . . . . . 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 863	. . . . . . . 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 838	. . . . . . 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 602	. . . . 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 1716	. . . 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 833	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 777   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814  <.cop 2415   class class class wbr 2624  ` cfv 3188  (class class class)co 3969  CCcc 5244   + caddc 5249   x. cmul 5251  ZZcz 5310  ZZ>cuz 6418  ...cfz 6468   seq cseqz 6532   ~~> cli 6974

This theorem is referenced by:  isummulc1 7212  ef1tllem 7381  ef01tllem1 7383

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim