MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lmcn2 Unicode version

Theorem lmcn2 17343
Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 15-May-2014.)
Hypotheses
Ref Expression
txlm.z  |-  Z  =  ( ZZ>= `  M )
txlm.m  |-  ( ph  ->  M  e.  ZZ )
txlm.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
txlm.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
txlm.f  |-  ( ph  ->  F : Z --> X )
txlm.g  |-  ( ph  ->  G : Z --> Y )
lmcn2.fl  |-  ( ph  ->  F ( ~~> t `  J ) R )
lmcn2.gl  |-  ( ph  ->  G ( ~~> t `  K ) S )
lmcn2.o  |-  ( ph  ->  O  e.  ( ( J  tX  K )  Cn  N ) )
lmcn2.h  |-  H  =  ( n  e.  Z  |->  ( ( F `  n ) O ( G `  n ) ) )
Assertion
Ref Expression
lmcn2  |-  ( ph  ->  H ( ~~> t `  N ) ( R O S ) )
Distinct variable groups:    n, F    n, O    ph, n    n, G    n, J    n, K    n, X    n, Y    n, Z
Allowed substitution hints:    R( n)    S( n)    H( n)    M( n)    N( n)

Proof of Theorem lmcn2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 txlm.f . . . . . . 7  |-  ( ph  ->  F : Z --> X )
2 ffvelrn 5663 . . . . . . 7  |-  ( ( F : Z --> X  /\  n  e.  Z )  ->  ( F `  n
)  e.  X )
31, 2sylan 457 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  X )
4 txlm.g . . . . . . 7  |-  ( ph  ->  G : Z --> Y )
5 ffvelrn 5663 . . . . . . 7  |-  ( ( G : Z --> Y  /\  n  e.  Z )  ->  ( G `  n
)  e.  Y )
64, 5sylan 457 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( G `  n )  e.  Y )
7 opelxpi 4721 . . . . . 6  |-  ( ( ( F `  n
)  e.  X  /\  ( G `  n )  e.  Y )  ->  <. ( F `  n
) ,  ( G `
 n ) >.  e.  ( X  X.  Y
) )
83, 6, 7syl2anc 642 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  <. ( F `  n ) ,  ( G `  n ) >.  e.  ( X  X.  Y ) )
9 eqidd 2284 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
<. ( F `  n
) ,  ( G `
 n ) >.
)  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )
)
10 txlm.j . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  X ) )
11 txlm.k . . . . . . . 8  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
12 txtopon 17286 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
1310, 11, 12syl2anc 642 . . . . . . 7  |-  ( ph  ->  ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) ) )
14 lmcn2.o . . . . . . . . 9  |-  ( ph  ->  O  e.  ( ( J  tX  K )  Cn  N ) )
15 cntop2 16971 . . . . . . . . 9  |-  ( O  e.  ( ( J 
tX  K )  Cn  N )  ->  N  e.  Top )
1614, 15syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
17 eqid 2283 . . . . . . . . 9  |-  U. N  =  U. N
1817toptopon 16671 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1916, 18sylib 188 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
20 cnf2 16979 . . . . . . 7  |-  ( ( ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) )  /\  N  e.  (TopOn `  U. N )  /\  O  e.  ( ( J  tX  K
)  Cn  N ) )  ->  O :
( X  X.  Y
) --> U. N )
2113, 19, 14, 20syl3anc 1182 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
2221feqmptd 5575 . . . . 5  |-  ( ph  ->  O  =  ( x  e.  ( X  X.  Y )  |->  ( O `
 x ) ) )
23 fveq2 5525 . . . . . 6  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
)
24 df-ov 5861 . . . . . 6  |-  ( ( F `  n ) O ( G `  n ) )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
2523, 24syl6eqr 2333 . . . . 5  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( ( F `  n
) O ( G `
 n ) ) )
268, 9, 22, 25fmptco 5691 . . . 4  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) )  =  ( n  e.  Z  |->  ( ( F `  n
) O ( G `
 n ) ) ) )
27 lmcn2.h . . . 4  |-  H  =  ( n  e.  Z  |->  ( ( F `  n ) O ( G `  n ) ) )
2826, 27syl6eqr 2333 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) )  =  H )
29 lmcn2.fl . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) R )
30 lmcn2.gl . . . . 5  |-  ( ph  ->  G ( ~~> t `  K ) S )
31 txlm.z . . . . . 6  |-  Z  =  ( ZZ>= `  M )
32 txlm.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
33 eqid 2283 . . . . . 6  |-  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n )
>. )
3431, 32, 10, 11, 1, 4, 33txlm 17342 . . . . 5  |-  ( ph  ->  ( ( F ( ~~> t `  J ) R  /\  G ( ~~> t `  K ) S )  <->  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )
( ~~> t `  ( J  tX  K ) )
<. R ,  S >. ) )
3529, 30, 34mpbi2and 887 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
<. ( F `  n
) ,  ( G `
 n ) >.
) ( ~~> t `  ( J  tX  K ) ) <. R ,  S >. )
3635, 14lmcn 17033 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) ) ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
3728, 36eqbrtrrd 4045 . 2  |-  ( ph  ->  H ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
38 df-ov 5861 . 2  |-  ( R O S )  =  ( O `  <. R ,  S >. )
3937, 38syl6breqr 4063 1  |-  ( ph  ->  H ( ~~> t `  N ) ( R O S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   ZZcz 10024   ZZ>=cuz 10230   Topctop 16631  TopOnctopon 16632    Cn ccn 16954   ~~> tclm 16956    tX ctx 17255
This theorem is referenced by:  hlimadd  21772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-z 10025  df-uz 10231  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cnp 16958  df-lm 16959  df-tx 17257
  Copyright terms: Public domain W3C validator