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Theorem climmul 11376
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 11331 . . 3  |-  ( F  ~~>  A  ->  A  e.  CC )
53, 4syl 14 . 2  |-  ( ph  ->  A  e.  CC )
6 climadd.7 . . 3  |-  ( ph  ->  G  ~~>  B )
7 climcl 11331 . . 3  |-  ( G  ~~>  B  ->  B  e.  CC )
86, 7syl 14 . 2  |-  ( ph  ->  B  e.  CC )
9 mulcl 7973 . . 3  |-  ( ( u  e.  CC  /\  v  e.  CC )  ->  ( u  x.  v
)  e.  CC )
109adantl 277 . 2  |-  ( (
ph  /\  ( u  e.  CC  /\  v  e.  CC ) )  -> 
( u  x.  v
)  e.  CC )
11 climadd.6 . 2  |-  ( ph  ->  H  e.  X )
12 simpr 110 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  x  e.  RR+ )
135adantr 276 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  A  e.  CC )
148adantr 276 . . 3  |-  ( (
ph  /\  x  e.  RR+ )  ->  B  e.  CC )
15 mulcn2 11361 . . 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 1249 . 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 11358 1  |-  ( ph  ->  H  ~~>  ( A  x.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   A.wral 2468   E.wrex 2469   class class class wbr 4021   ` cfv 5238  (class class class)co 5900   CCcc 7844    x. cmul 7851    < clt 8027    - cmin 8163   ZZcz 9288   ZZ>=cuz 9563   RR+crp 9689   abscabs 11047    ~~> cli 11327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4136  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-iinf 4608  ax-cnex 7937  ax-resscn 7938  ax-1cn 7939  ax-1re 7940  ax-icn 7941  ax-addcl 7942  ax-addrcl 7943  ax-mulcl 7944  ax-mulrcl 7945  ax-addcom 7946  ax-mulcom 7947  ax-addass 7948  ax-mulass 7949  ax-distr 7950  ax-i2m1 7951  ax-0lt1 7952  ax-1rid 7953  ax-0id 7954  ax-rnegex 7955  ax-precex 7956  ax-cnre 7957  ax-pre-ltirr 7958  ax-pre-ltwlin 7959  ax-pre-lttrn 7960  ax-pre-apti 7961  ax-pre-ltadd 7962  ax-pre-mulgt0 7963  ax-pre-mulext 7964  ax-arch 7965  ax-caucvg 7966
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-iun 3906  df-br 4022  df-opab 4083  df-mpt 4084  df-tr 4120  df-id 4314  df-po 4317  df-iso 4318  df-iord 4387  df-on 4389  df-ilim 4390  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-riota 5855  df-ov 5903  df-oprab 5904  df-mpo 5905  df-1st 6169  df-2nd 6170  df-recs 6334  df-frec 6420  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-sub 8165  df-neg 8166  df-reap 8567  df-ap 8574  df-div 8665  df-inn 8955  df-2 9013  df-3 9014  df-4 9015  df-n0 9212  df-z 9289  df-uz 9564  df-rp 9690  df-seqfrec 10485  df-exp 10560  df-cj 10892  df-re 10893  df-im 10894  df-rsqrt 11048  df-abs 11049  df-clim 11328
This theorem is referenced by:  climmulc2  11380
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