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Theorem climmul 11108
Description: Limit of the product of two converging sequences. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by NM, 27-Dec-2005.) (Proof shortened by Mario Carneiro, 1-Feb-2014.)
Hypotheses
Ref Expression
climadd.1  |-  Z  =  ( ZZ>= `  M )
climadd.2  |-  ( ph  ->  M  e.  ZZ )
climadd.4  |-  ( ph  ->  F  ~~>  A )
climadd.6  |-  ( ph  ->  H  e.  X )
climadd.7  |-  ( ph  ->  G  ~~>  B )
climadd.8  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
climadd.9  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
climmul.h  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
Assertion
Ref Expression
climmul  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Distinct variable groups:    B, k    k, F    ph, k    A, k   
k, G    k, H    k, M    k, Z
Allowed substitution hint:    X( k)

Proof of Theorem climmul
Dummy variables  u  v  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 climadd.1 . 2  |-  Z  =  ( ZZ>= `  M )
2 climadd.2 . 2  |-  ( ph  ->  M  e.  ZZ )
3 climadd.4 . . 3  |-  ( ph  ->  F  ~~>  A )
4 climcl 11063 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 14 . 2  |-  ( ph  ->  A  e.  CC )
6 climadd.7 . . 3  |-  ( ph  ->  G  ~~>  B )
7 climcl 11063 . . 3  |-  ( G  ~~>  B  ->  B  e.  CC )
86, 7syl 14 . 2  |-  ( ph  ->  B  e.  CC )
9 mulcl 7759 . . 3  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  x.  v
)  e.  CC )
109adantl 275 . 2  |-  ( (
ph  /\  ( u  e.  CC  /\  v  e.  CC ) )  -> 
( u  x.  v
)  e.  CC )
11 climadd.6 . 2  |-  ( ph  ->  H  e.  X )
12 simpr 109 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
135adantr 274 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  A  e.  CC )
148adantr 274 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  B  e.  CC )
15 mulcn2 11093 . . 3  |-  ( ( x  e.  RR+  /\  A  e.  CC  /\  B  e.  CC )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  A ) )  <  y  /\  ( abs `  ( v  -  B ) )  <  z )  -> 
( abs `  (
( u  x.  v
)  -  ( A  x.  B ) ) )  <  x ) )
1612, 13, 14, 15syl3anc 1216 . 2  |-  ( (
ph  /\  x  e.  RR+ )  ->  E. y  e.  RR+  E. z  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  A ) )  <  y  /\  ( abs `  ( v  -  B ) )  <  z )  -> 
( abs `  (
( u  x.  v
)  -  ( A  x.  B ) ) )  <  x ) )
17 climadd.8 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( F `  k )  e.  CC )
18 climadd.9 . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( G `  k )  e.  CC )
19 climmul.h . 2  |-  ( (
ph  /\  k  e.  Z )  ->  ( H `  k )  =  ( ( F `
 k )  x.  ( G `  k
) ) )
201, 2, 5, 8, 10, 3, 6, 11, 16, 17, 18, 19climcn2 11090 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630    x. cmul 7637    < clt 7812    - cmin 7945   ZZcz 9066   ZZ>=cuz 9338   RR+crp 9453   abscabs 10781    ~~> cli 11059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-rp 9454  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783  df-clim 11060
This theorem is referenced by:  climmulc2  11112
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