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Theorem lmcn2 20741
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 )
21ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  X )
3 txlm.g . . . . . . 7  |-  ( ph  ->  G : Z --> Y )
43ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( G `  n )  e.  Y )
5 opelxpi 4871 . . . . . 6  |-  ( ( ( F `  n
)  e.  X  /\  ( G `  n )  e.  Y )  ->  <. ( F `  n
) ,  ( G `
 n ) >.  e.  ( X  X.  Y
) )
62, 4, 5syl2anc 673 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  <. ( F `  n ) ,  ( G `  n ) >.  e.  ( X  X.  Y ) )
7 eqidd 2472 . . . . 5  |-  ( ph  ->  ( n  e.  Z  |-> 
<. ( F `  n
) ,  ( G `
 n ) >.
)  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )
)
8 txlm.j . . . . . . . 8  |-  ( ph  ->  J  e.  (TopOn `  X ) )
9 txlm.k . . . . . . . 8  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
10 txtopon 20683 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
118, 9, 10syl2anc 673 . . . . . . 7  |-  ( ph  ->  ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) ) )
12 lmcn2.o . . . . . . . . 9  |-  ( ph  ->  O  e.  ( ( J  tX  K )  Cn  N ) )
13 cntop2 20334 . . . . . . . . 9  |-  ( O  e.  ( ( J 
tX  K )  Cn  N )  ->  N  e.  Top )
1412, 13syl 17 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
15 eqid 2471 . . . . . . . . 9  |-  U. N  =  U. N
1615toptopon 20025 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1714, 16sylib 201 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
18 cnf2 20342 . . . . . . 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 )
1911, 17, 12, 18syl3anc 1292 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
2019feqmptd 5932 . . . . 5  |-  ( ph  ->  O  =  ( x  e.  ( X  X.  Y )  |->  ( O `
 x ) ) )
21 fveq2 5879 . . . . . 6  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
)
22 df-ov 6311 . . . . . 6  |-  ( ( F `  n ) O ( G `  n ) )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
2321, 22syl6eqr 2523 . . . . 5  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( ( F `  n
) O ( G `
 n ) ) )
246, 7, 20, 23fmptco 6072 . . . 4  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) )  =  ( n  e.  Z  |->  ( ( F `  n
) O ( G `
 n ) ) ) )
25 lmcn2.h . . . 4  |-  H  =  ( n  e.  Z  |->  ( ( F `  n ) O ( G `  n ) ) )
2624, 25syl6eqr 2523 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) )  =  H )
27 lmcn2.fl . . . . 5  |-  ( ph  ->  F ( ~~> t `  J ) R )
28 lmcn2.gl . . . . 5  |-  ( ph  ->  G ( ~~> t `  K ) S )
29 txlm.z . . . . . 6  |-  Z  =  ( ZZ>= `  M )
30 txlm.m . . . . . 6  |-  ( ph  ->  M  e.  ZZ )
31 eqid 2471 . . . . . 6  |-  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n )
>. )
3229, 30, 8, 9, 1, 3, 31txlm 20740 . . . . 5  |-  ( ph  ->  ( ( F ( ~~> t `  J ) R  /\  G ( ~~> t `  K ) S )  <->  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )
( ~~> t `  ( J  tX  K ) )
<. R ,  S >. ) )
3327, 28, 32mpbi2and 935 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
<. ( F `  n
) ,  ( G `
 n ) >.
) ( ~~> t `  ( J  tX  K ) ) <. R ,  S >. )
3433, 12lmcn 20398 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) ) ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
3526, 34eqbrtrrd 4418 . 2  |-  ( ph  ->  H ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
36 df-ov 6311 . 2  |-  ( R O S )  =  ( O `  <. R ,  S >. )
3735, 36syl6breqr 4436 1  |-  ( ph  ->  H ( ~~> t `  N ) ( R O S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   <.cop 3965   U.cuni 4190   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837    o. ccom 4843   -->wf 5585   ` cfv 5589  (class class class)co 6308   ZZcz 10961   ZZ>=cuz 11182   Topctop 19994  TopOnctopon 19995    Cn ccn 20317   ~~> tclm 20319    tX ctx 20652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-pm 7493  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-z 10962  df-uz 11183  df-topgen 15420  df-top 19998  df-bases 19999  df-topon 20000  df-cn 20320  df-cnp 20321  df-lm 20322  df-tx 20654
This theorem is referenced by:  hlimadd  26927
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