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Theorem rlimmul 12112
Description: Limit of the product of two converging functions. Proposition 12-2.1(c) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.)
Hypotheses
Ref Expression
rlimadd.3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
rlimadd.4  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  V )
rlimadd.5  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D
)
rlimadd.6  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E
)
Assertion
Ref Expression
rlimmul  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C
) )  ~~> r  ( D  x.  E ) )
Distinct variable groups:    x, A    x, D    ph, x    x, E
Dummy variables  w  v  y  z  u are mutually distinct and distinct from all other variables.
Allowed substitution hints:    B( x)    C( x)    V( x)

Proof of Theorem rlimmul
StepHypRef Expression
1 rlimadd.3 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  V )
2 rlimadd.5 . . 3  |-  ( ph  ->  ( x  e.  A  |->  B )  ~~> r  D
)
31, 2rlimmptrcl 12075 . 2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  CC )
4 rlimadd.4 . . 3  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  V )
5 rlimadd.6 . . 3  |-  ( ph  ->  ( x  e.  A  |->  C )  ~~> r  E
)
64, 5rlimmptrcl 12075 . 2  |-  ( (
ph  /\  x  e.  A )  ->  C  e.  CC )
7 rlimcl 11971 . . 3  |-  ( ( x  e.  A  |->  B )  ~~> r  D  ->  D  e.  CC )
82, 7syl 17 . 2  |-  ( ph  ->  D  e.  CC )
9 rlimcl 11971 . . 3  |-  ( ( x  e.  A  |->  C )  ~~> r  E  ->  E  e.  CC )
105, 9syl 17 . 2  |-  ( ph  ->  E  e.  CC )
11 ax-mulf 8812 . . 3  |-  x.  :
( CC  X.  CC )
--> CC
1211a1i 12 . 2  |-  ( ph  ->  x.  : ( CC 
X.  CC ) --> CC )
13 simpr 449 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  y  e.  RR+ )
148adantr 453 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  D  e.  CC )
1510adantr 453 . . 3  |-  ( (
ph  /\  y  e.  RR+ )  ->  E  e.  CC )
16 mulcn2 12063 . . 3  |-  ( ( y  e.  RR+  /\  D  e.  CC  /\  E  e.  CC )  ->  E. z  e.  RR+  E. w  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  D ) )  <  z  /\  ( abs `  ( v  -  E ) )  <  w )  -> 
( abs `  (
( u  x.  v
)  -  ( D  x.  E ) ) )  <  y ) )
1713, 14, 15, 16syl3anc 1184 . 2  |-  ( (
ph  /\  y  e.  RR+ )  ->  E. z  e.  RR+  E. w  e.  RR+  A. u  e.  CC  A. v  e.  CC  (
( ( abs `  (
u  -  D ) )  <  z  /\  ( abs `  ( v  -  E ) )  <  w )  -> 
( abs `  (
( u  x.  v
)  -  ( D  x.  E ) ) )  <  y ) )
183, 6, 8, 10, 2, 5, 12, 17rlimcn2 12058 1  |-  ( ph  ->  ( x  e.  A  |->  ( B  x.  C
) )  ~~> r  ( D  x.  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    e. wcel 1685   A.wral 2544   E.wrex 2545   class class class wbr 4024    e. cmpt 4078    X. cxp 4686   -->wf 5217   ` cfv 5221  (class class class)co 5819   CCcc 8730    x. cmul 8737    < clt 8862    - cmin 9032   RR+crp 10349   abscabs 11713    ~~> r crli 11953
This theorem is referenced by:  rlimdiv  12113  caucvgr  12142  logexprlim  20458  dchrisum0lem1  20659
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809  ax-pre-sup 8810  ax-mulf 8812
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  df-ral 2549  df-rex 2550  df-reu 2551  df-rmo 2552  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-pm 6770  df-en 6859  df-dom 6860  df-sdom 6861  df-sup 7189  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-div 9419  df-nn 9742  df-2 9799  df-3 9800  df-n0 9961  df-z 10020  df-uz 10226  df-rp 10350  df-seq 11041  df-exp 11099  df-cj 11578  df-re 11579  df-im 11580  df-sqr 11714  df-abs 11715  df-rlim 11957
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