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Theorem lmcn2 17359
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 5679 . . . . . . 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 5679 . . . . . . 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 4737 . . . . . 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 2297 . . . . 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 17302 . . . . . . . 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 16987 . . . . . . . . 9  |-  ( O  e.  ( ( J 
tX  K )  Cn  N )  ->  N  e.  Top )
1614, 15syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
17 eqid 2296 . . . . . . . . 9  |-  U. N  =  U. N
1817toptopon 16687 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1916, 18sylib 188 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
20 cnf2 16995 . . . . . . 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 5591 . . . . 5  |-  ( ph  ->  O  =  ( x  e.  ( X  X.  Y )  |->  ( O `
 x ) ) )
23 fveq2 5541 . . . . . 6  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
)
24 df-ov 5877 . . . . . 6  |-  ( ( F `  n ) O ( G `  n ) )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
2523, 24syl6eqr 2346 . . . . 5  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( ( F `  n
) O ( G `
 n ) ) )
268, 9, 22, 25fmptco 5707 . . . 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 2346 . . 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 2296 . . . . . 6  |-  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n )
>. )
3431, 32, 10, 11, 1, 4, 33txlm 17358 . . . . 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 17049 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) ) ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
3728, 36eqbrtrrd 4061 . 2  |-  ( ph  ->  H ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
38 df-ov 5877 . 2  |-  ( R O S )  =  ( O `  <. R ,  S >. )
3937, 38syl6breqr 4079 1  |-  ( ph  ->  H ( ~~> t `  N ) ( R O S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   <.cop 3656   U.cuni 3843   class class class wbr 4039    e. cmpt 4093    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   ZZcz 10040   ZZ>=cuz 10246   Topctop 16647  TopOnctopon 16648    Cn ccn 16970   ~~> tclm 16972    tX ctx 17271
This theorem is referenced by:  hlimadd  21788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-z 10041  df-uz 10247  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-cnp 16974  df-lm 16975  df-tx 17273
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