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Theorem climmul 12381
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 12248 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 16 . 2  |-  ( ph  ->  A  e.  CC )
6 climadd.7 . . 3  |-  ( ph  ->  G  ~~>  B )
7 climcl 12248 . . 3  |-  ( G  ~~>  B  ->  B  e.  CC )
86, 7syl 16 . 2  |-  ( ph  ->  B  e.  CC )
9 mulcl 9030 . . 3  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  x.  v
)  e.  CC )
109adantl 453 . 2  |-  ( (
ph  /\  ( u  e.  CC  /\  v  e.  CC ) )  -> 
( u  x.  v
)  e.  CC )
11 climadd.6 . 2  |-  ( ph  ->  H  e.  X )
12 simpr 448 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
135adantr 452 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  A  e.  CC )
148adantr 452 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  B  e.  CC )
15 mulcn2 12344 . . 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 1184 . 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 12341 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   E.wrex 2667   class class class wbr 4172   ` cfv 5413  (class class class)co 6040   CCcc 8944    x. cmul 8951    < clt 9076    - cmin 9247   ZZcz 10238   ZZ>=cuz 10444   RR+crp 10568   abscabs 11994    ~~> cli 12233
This theorem is referenced by:  climmulc2  12385  mbfmullem2  19569  basellem7  20822  basellem9  20824  ntrivcvgmullem  25182  iprodmul  25269  faclim  25313  faclim2  25315  climmulf  27597
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-sup 7404  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-seq 11279  df-exp 11338  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237
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