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Theorem flfcnp2 20376
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 6298 . 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 19960 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
52, 3, 4syl2anc 661 . . . 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 5037 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
107, 8, 9syl2anc 661 . . . . 5  |-  ( (
ph  /\  x  e.  Z )  ->  <. A ,  B >.  e.  ( X  X.  Y ) )
11 eqid 2467 . . . . 5  |-  ( x  e.  Z  |->  <. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. )
1210, 11fmptd 6056 . . . 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 2467 . . . . . . . 8  |-  ( x  e.  Z  |->  A )  =  ( x  e.  Z  |->  A )
167, 15fmptd 6056 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  A ) : Z --> X )
17 eqid 2467 . . . . . . . 8  |-  ( x  e.  Z  |->  B )  =  ( x  e.  Z  |->  B )
188, 17fmptd 6056 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  B ) : Z --> Y )
19 nfcv 2629 . . . . . . . 8  |-  F/_ y <. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >.
20 nffvmpt1 5880 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  A ) `  y )
21 nffvmpt1 5880 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  B ) `  y )
2220, 21nfop 4235 . . . . . . . 8  |-  F/_ x <. ( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `  y
) >.
23 fveq2 5872 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  A ) `  x
)  =  ( ( x  e.  Z  |->  A ) `  y ) )
24 fveq2 5872 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  B ) `  x
)  =  ( ( x  e.  Z  |->  B ) `  y ) )
2523, 24opeq12d 4227 . . . . . . . 8  |-  ( x  =  y  ->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>.  =  <. ( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `
 y ) >.
)
2619, 22, 25cbvmpt 4543 . . . . . . 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
) >. )
272, 3, 6, 16, 18, 26txflf 20375 . . . . . 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 ) ) ) ) )
2813, 14, 27mpbir2and 920 . . . . 5  |-  ( ph  -> 
<. R ,  S >.  e.  ( ( ( J 
tX  K )  fLimf  L ) `  ( x  e.  Z  |->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>. ) ) )
29 simpr 461 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Z )  ->  x  e.  Z )
3015fvmpt2 5964 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  A  e.  X )  ->  ( ( x  e.  Z  |->  A ) `  x )  =  A )
3129, 7, 30syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  A ) `  x
)  =  A )
3217fvmpt2 5964 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  B  e.  Y )  ->  ( ( x  e.  Z  |->  B ) `  x )  =  B )
3329, 8, 32syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  B ) `  x
)  =  B )
3431, 33opeq12d 4227 . . . . . . 7  |-  ( (
ph  /\  x  e.  Z )  ->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>.  =  <. A ,  B >. )
3534mpteq2dva 4539 . . . . . 6  |-  ( ph  ->  ( x  e.  Z  |-> 
<. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
3635fveq2d 5876 . . . . 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 >. ) ) )
3728, 36eleqtrd 2557 . . . 4  |-  ( ph  -> 
<. R ,  S >.  e.  ( ( ( J 
tX  K )  fLimf  L ) `  ( x  e.  Z  |->  <. A ,  B >. ) ) )
38 flfcnp2.o . . . 4  |-  ( ph  ->  O  e.  ( ( ( J  tX  K
)  CnP  N ) `  <. R ,  S >. ) )
39 flfcnp 20373 . . . 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 >. ) ) ) )
405, 6, 12, 37, 38, 39syl32anc 1236 . . 3  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) ) )
41 eqidd 2468 . . . . 5  |-  ( ph  ->  ( x  e.  Z  |-> 
<. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
42 cnptop2 19612 . . . . . . . . 9  |-  ( O  e.  ( ( ( J  tX  K )  CnP  N ) `  <. R ,  S >. )  ->  N  e.  Top )
4338, 42syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
44 eqid 2467 . . . . . . . . 9  |-  U. N  =  U. N
4544toptopon 19303 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
4643, 45sylib 196 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
47 cnpf2 19619 . . . . . . 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
)
485, 46, 38, 47syl3anc 1228 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
4948feqmptd 5927 . . . . 5  |-  ( ph  ->  O  =  ( y  e.  ( X  X.  Y )  |->  ( O `
 y ) ) )
50 fveq2 5872 . . . . . 6  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( O `  <. A ,  B >. ) )
51 df-ov 6298 . . . . . 6  |-  ( A O B )  =  ( O `  <. A ,  B >. )
5250, 51syl6eqr 2526 . . . . 5  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( A O B ) )
5310, 41, 49, 52fmptco 6065 . . . 4  |-  ( ph  ->  ( O  o.  (
x  e.  Z  |->  <. A ,  B >. ) )  =  ( x  e.  Z  |->  ( A O B ) ) )
5453fveq2d 5876 . . 3  |-  ( ph  ->  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) )  =  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
5540, 54eleqtrd 2557 . 2  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
561, 55syl5eqel 2559 1  |-  ( ph  ->  ( R O S )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   <.cop 4039   U.cuni 4251    |-> cmpt 4511    X. cxp 5003    o. ccom 5009   -->wf 5590   ` cfv 5594  (class class class)co 6295   Topctop 19263  TopOnctopon 19264    CnP ccnp 19594    tX ctx 19929   Filcfil 20214    fLimf cflf 20304
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-map 7434  df-topgen 14716  df-fbas 18286  df-fg 18287  df-top 19268  df-bases 19270  df-topon 19271  df-ntr 19389  df-nei 19467  df-cnp 19597  df-tx 19931  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309
This theorem is referenced by:  tsmsadd  20517
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