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Theorem iisermulc2 10692
Description: Multiplication of an infinite series by a constant. New proofs should use isermulc2 10693 instead. (Contributed by Paul Chapman, 14-Nov-2007.) (Revised by Mario Carneiro, 1-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
clim2iser.1  |-  Z  =  ( ZZ>= `  M )
iisermulc2.2  |-  ( ph  ->  M  e.  ZZ )
iisermulc2.4  |-  ( ph  ->  C  e.  CC )
iisermulc2.5  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
iisermulc2.6  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
iisermulc2.7  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )
Assertion
Ref Expression
iisermulc2  |-  ( ph  ->  seq M (  +  ,  G ,  CC ) 
~~>  ( C  x.  A
) )
Distinct variable groups:    A, k    k, F    k, M    C, k    k, G    ph, k    k, Z

Proof of Theorem iisermulc2
Dummy variables  j  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clim2iser.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 iisermulc2.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 iisermulc2.5 . 2  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) 
~~>  A )
4 iisermulc2.4 . 2  |-  ( ph  ->  C  e.  CC )
5 iseqex 9821 . . 3  |-  seq M
(  +  ,  G ,  CC )  e.  _V
65a1i 9 . 2  |-  ( ph  ->  seq M (  +  ,  G ,  CC )  e.  _V )
7 iisermulc2.6 . . . 4  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
81, 2, 7iserf 9868 . . 3  |-  ( ph  ->  seq M (  +  ,  F ,  CC ) : Z --> CC )
98ffvelrnda 5418 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  F ,  CC ) `  j )  e.  CC )
10 addcl 7446 . . . 4  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  +  x
)  e.  CC )
1110adantl 271 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  +  x )  e.  CC )
124adantr 270 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  C  e.  CC )
13 adddi 7453 . . . . 5  |-  ( ( C  e.  CC  /\  k  e.  CC  /\  x  e.  CC )  ->  ( C  x.  ( k  +  x ) )  =  ( ( C  x.  k )  +  ( C  x.  x ) ) )
14133expb 1144 . . . 4  |-  ( ( C  e.  CC  /\  ( k  e.  CC  /\  x  e.  CC ) )  ->  ( C  x.  ( k  +  x
) )  =  ( ( C  x.  k
)  +  ( C  x.  x ) ) )
1512, 14sylan 277 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( C  x.  ( k  +  x
) )  =  ( ( C  x.  k
)  +  ( C  x.  x ) ) )
16 simpr 108 . . . 4  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  Z )
1716, 1syl6eleq 2180 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  j  e.  ( ZZ>= `  M )
)
181eleq2i 2154 . . . . 5  |-  ( k  e.  Z  <->  k  e.  ( ZZ>= `  M )
)
1918, 7sylan2br 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
2019adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( F `  k )  e.  CC )
21 iisermulc2.7 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  =  ( C  x.  ( F `  k ) ) )
2218, 21sylan2br 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( C  x.  ( F `
 k ) ) )
2322adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  =  ( C  x.  ( F `
 k ) ) )
24 cnex 7445 . . . 4  |-  CC  e.  _V
2524a1i 9 . . 3  |-  ( (
ph  /\  j  e.  Z )  ->  CC  e.  _V )
26 mulcl 7448 . . . 4  |-  ( ( k  e.  CC  /\  x  e.  CC )  ->  ( k  x.  x
)  e.  CC )
2726adantl 271 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  (
k  e.  CC  /\  x  e.  CC )
)  ->  ( k  x.  x )  e.  CC )
284adantr 270 . . . . . . 7  |-  ( (
ph  /\  k  e.  Z )  ->  C  e.  CC )
2928, 7mulcld 7487 . . . . . 6  |-  ( (
ph  /\  k  e.  Z )  ->  ( C  x.  ( F `  k ) )  e.  CC )
3021, 29eqeltrd 2164 . . . . 5  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
3118, 30sylan2br 282 . . . 4  |-  ( (
ph  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3231adantlr 461 . . 3  |-  ( ( ( ph  /\  j  e.  Z )  /\  k  e.  ( ZZ>= `  M )
)  ->  ( G `  k )  e.  CC )
3311, 15, 17, 20, 23, 25, 27, 32, 12iseqdistr 9910 . 2  |-  ( (
ph  /\  j  e.  Z )  ->  (  seq M (  +  ,  G ,  CC ) `  j )  =  ( C  x.  (  seq M (  +  ,  F ,  CC ) `  j ) ) )
341, 2, 3, 4, 6, 9, 33climmulc2 10683 1  |-  ( ph  ->  seq M (  +  ,  G ,  CC ) 
~~>  ( C  x.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438   _Vcvv 2619   class class class wbr 3837   ` cfv 5002  (class class class)co 5634   CCcc 7327    + caddc 7332    x. cmul 7334   ZZcz 8720   ZZ>=cuz 8988    seqcseq4 9816    ~~> cli 10630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-nul 3957  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-iinf 4393  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442  ax-arch 7443  ax-caucvg 7444
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-if 3390  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-po 4114  df-iso 4115  df-iord 4184  df-on 4186  df-ilim 4187  df-suc 4189  df-iom 4396  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-1st 5893  df-2nd 5894  df-recs 6052  df-frec 6138  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114  df-inn 8395  df-2 8452  df-3 8453  df-4 8454  df-n0 8644  df-z 8721  df-uz 8989  df-rp 9104  df-iseq 9818  df-seq3 9819  df-exp 9920  df-cj 10241  df-re 10242  df-im 10243  df-rsqrt 10396  df-abs 10397  df-clim 10631
This theorem is referenced by:  isermulc2  10693  isummulc2  10783
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