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Theorem flfcnp2 17702
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, 19-Sep-2015.)
Hypotheses
Ref Expression
flfcnp2.j  |-  ( ph  ->  J  e.  (TopOn `  X ) )
flfcnp2.k  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
flfcnp2.l  |-  ( ph  ->  L  e.  ( Fil `  Z ) )
flfcnp2.a  |-  ( (
ph  /\  x  e.  Z )  ->  A  e.  X )
flfcnp2.b  |-  ( (
ph  /\  x  e.  Z )  ->  B  e.  Y )
flfcnp2.r  |-  ( ph  ->  R  e.  ( ( J  fLimf  L ) `  ( x  e.  Z  |->  A ) ) )
flfcnp2.s  |-  ( ph  ->  S  e.  ( ( K  fLimf  L ) `  ( x  e.  Z  |->  B ) ) )
flfcnp2.o  |-  ( ph  ->  O  e.  ( ( ( J  tX  K
)  CnP  N ) `  <. R ,  S >. ) )
Assertion
Ref Expression
flfcnp2  |-  ( ph  ->  ( R O S )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
Distinct variable groups:    x, O    ph, x    x, Z    x, X    x, Y
Allowed substitution hints:    A( x)    B( x)    R( x)    S( x)    J( x)    K( x)    L( x)    N( x)

Proof of Theorem flfcnp2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-ov 5861 . 2  |-  ( R O S )  =  ( O `  <. R ,  S >. )
2 flfcnp2.j . . . . 5  |-  ( ph  ->  J  e.  (TopOn `  X ) )
3 flfcnp2.k . . . . 5  |-  ( ph  ->  K  e.  (TopOn `  Y ) )
4 txtopon 17286 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
52, 3, 4syl2anc 642 . . . 4  |-  ( ph  ->  ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) ) )
6 flfcnp2.l . . . 4  |-  ( ph  ->  L  e.  ( Fil `  Z ) )
7 flfcnp2.a . . . . . 6  |-  ( (
ph  /\  x  e.  Z )  ->  A  e.  X )
8 flfcnp2.b . . . . . 6  |-  ( (
ph  /\  x  e.  Z )  ->  B  e.  Y )
9 opelxpi 4721 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
107, 8, 9syl2anc 642 . . . . 5  |-  ( (
ph  /\  x  e.  Z )  ->  <. A ,  B >.  e.  ( X  X.  Y ) )
11 eqid 2283 . . . . 5  |-  ( x  e.  Z  |->  <. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. )
1210, 11fmptd 5684 . . . 4  |-  ( ph  ->  ( x  e.  Z  |-> 
<. A ,  B >. ) : Z --> ( X  X.  Y ) )
13 flfcnp2.r . . . . . 6  |-  ( ph  ->  R  e.  ( ( J  fLimf  L ) `  ( x  e.  Z  |->  A ) ) )
14 flfcnp2.s . . . . . 6  |-  ( ph  ->  S  e.  ( ( K  fLimf  L ) `  ( x  e.  Z  |->  B ) ) )
15 eqid 2283 . . . . . . . 8  |-  ( x  e.  Z  |->  A )  =  ( x  e.  Z  |->  A )
167, 15fmptd 5684 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  A ) : Z --> X )
17 eqid 2283 . . . . . . . 8  |-  ( x  e.  Z  |->  B )  =  ( x  e.  Z  |->  B )
188, 17fmptd 5684 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  B ) : Z --> Y )
19 nfcv 2419 . . . . . . . 8  |-  F/_ y <. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >.
20 nfmpt1 4109 . . . . . . . . . 10  |-  F/_ x
( x  e.  Z  |->  A )
21 nfcv 2419 . . . . . . . . . 10  |-  F/_ x
y
2220, 21nffv 5532 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  A ) `  y )
23 nfmpt1 4109 . . . . . . . . . 10  |-  F/_ x
( x  e.  Z  |->  B )
2423, 21nffv 5532 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  B ) `  y )
2522, 24nfop 3812 . . . . . . . 8  |-  F/_ x <. ( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `  y
) >.
26 fveq2 5525 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  A ) `  x
)  =  ( ( x  e.  Z  |->  A ) `  y ) )
27 fveq2 5525 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  B ) `  x
)  =  ( ( x  e.  Z  |->  B ) `  y ) )
2826, 27opeq12d 3804 . . . . . . . 8  |-  ( x  =  y  ->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>.  =  <. ( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `
 y ) >.
)
2919, 25, 28cbvmpt 4110 . . . . . . 7  |-  ( x  e.  Z  |->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>. )  =  (
y  e.  Z  |->  <.
( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `  y
) >. )
302, 3, 6, 16, 18, 29txflf 17701 . . . . . 6  |-  ( ph  ->  ( <. R ,  S >.  e.  ( ( ( J  tX  K ) 
fLimf  L ) `  (
x  e.  Z  |->  <.
( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >. ) )  <->  ( R  e.  ( ( J  fLimf  L ) `  ( x  e.  Z  |->  A ) )  /\  S  e.  ( ( K  fLimf  L ) `  ( x  e.  Z  |->  B ) ) ) ) )
3113, 14, 30mpbir2and 888 . . . . 5  |-  ( ph  -> 
<. R ,  S >.  e.  ( ( ( J 
tX  K )  fLimf  L ) `  ( x  e.  Z  |->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>. ) ) )
32 simpr 447 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Z )  ->  x  e.  Z )
3315fvmpt2 5608 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  A  e.  X )  ->  ( ( x  e.  Z  |->  A ) `  x )  =  A )
3432, 7, 33syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  A ) `  x
)  =  A )
3517fvmpt2 5608 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  B  e.  Y )  ->  ( ( x  e.  Z  |->  B ) `  x )  =  B )
3632, 8, 35syl2anc 642 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  B ) `  x
)  =  B )
3734, 36opeq12d 3804 . . . . . . 7  |-  ( (
ph  /\  x  e.  Z )  ->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>.  =  <. A ,  B >. )
3837mpteq2dva 4106 . . . . . 6  |-  ( ph  ->  ( x  e.  Z  |-> 
<. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
3938fveq2d 5529 . . . . 5  |-  ( ph  ->  ( ( ( J 
tX  K )  fLimf  L ) `  ( x  e.  Z  |->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>. ) )  =  ( ( ( J  tX  K )  fLimf  L ) `
 ( x  e.  Z  |->  <. A ,  B >. ) ) )
4031, 39eleqtrd 2359 . . . 4  |-  ( ph  -> 
<. R ,  S >.  e.  ( ( ( J 
tX  K )  fLimf  L ) `  ( x  e.  Z  |->  <. A ,  B >. ) ) )
41 flfcnp2.o . . . 4  |-  ( ph  ->  O  e.  ( ( ( J  tX  K
)  CnP  N ) `  <. R ,  S >. ) )
42 flfcnp 17699 . . . 4  |-  ( ( ( ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) )  /\  L  e.  ( Fil `  Z
)  /\  ( x  e.  Z  |->  <. A ,  B >. ) : Z --> ( X  X.  Y
) )  /\  ( <. R ,  S >.  e.  ( ( ( J 
tX  K )  fLimf  L ) `  ( x  e.  Z  |->  <. A ,  B >. ) )  /\  O  e.  ( (
( J  tX  K
)  CnP  N ) `  <. R ,  S >. ) ) )  -> 
( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) ) )
435, 6, 12, 40, 41, 42syl32anc 1190 . . 3  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) ) )
44 eqidd 2284 . . . . 5  |-  ( ph  ->  ( x  e.  Z  |-> 
<. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
45 cnptop2 16973 . . . . . . . . 9  |-  ( O  e.  ( ( ( J  tX  K )  CnP  N ) `  <. R ,  S >. )  ->  N  e.  Top )
4641, 45syl 15 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
47 eqid 2283 . . . . . . . . 9  |-  U. N  =  U. N
4847toptopon 16671 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
4946, 48sylib 188 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
50 cnpf2 16980 . . . . . . 7  |-  ( ( ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) )  /\  N  e.  (TopOn `  U. N )  /\  O  e.  ( ( ( J  tX  K )  CnP  N
) `  <. R ,  S >. ) )  ->  O : ( X  X.  Y ) --> U. N
)
515, 49, 41, 50syl3anc 1182 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
5251feqmptd 5575 . . . . 5  |-  ( ph  ->  O  =  ( y  e.  ( X  X.  Y )  |->  ( O `
 y ) ) )
53 fveq2 5525 . . . . . 6  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( O `  <. A ,  B >. ) )
54 df-ov 5861 . . . . . 6  |-  ( A O B )  =  ( O `  <. A ,  B >. )
5553, 54syl6eqr 2333 . . . . 5  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( A O B ) )
5610, 44, 52, 55fmptco 5691 . . . 4  |-  ( ph  ->  ( O  o.  (
x  e.  Z  |->  <. A ,  B >. ) )  =  ( x  e.  Z  |->  ( A O B ) ) )
5756fveq2d 5529 . . 3  |-  ( ph  ->  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) )  =  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
5843, 57eleqtrd 2359 . 2  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
591, 58syl5eqel 2367 1  |-  ( ph  ->  ( R O S )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   U.cuni 3827    e. cmpt 4077    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632    CnP ccnp 16955    tX ctx 17255   Filcfil 17540    fLimf cflf 17630
This theorem is referenced by:  tsmsadd  17829
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-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-ntr 16757  df-nei 16835  df-cnp 16958  df-tx 17257  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635
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