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Theorem flfcnp2 18000
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 6051 . 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 17584 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
52, 3, 4syl2anc 643 . . . 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 4877 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
107, 8, 9syl2anc 643 . . . . 5  |-  ( (
ph  /\  x  e.  Z )  ->  <. A ,  B >.  e.  ( X  X.  Y ) )
11 eqid 2412 . . . . 5  |-  ( x  e.  Z  |->  <. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. )
1210, 11fmptd 5860 . . . 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 2412 . . . . . . . 8  |-  ( x  e.  Z  |->  A )  =  ( x  e.  Z  |->  A )
167, 15fmptd 5860 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  A ) : Z --> X )
17 eqid 2412 . . . . . . . 8  |-  ( x  e.  Z  |->  B )  =  ( x  e.  Z  |->  B )
188, 17fmptd 5860 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  B ) : Z --> Y )
19 nfcv 2548 . . . . . . . 8  |-  F/_ y <. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >.
20 nffvmpt1 5703 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  A ) `  y )
21 nffvmpt1 5703 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  B ) `  y )
2220, 21nfop 3968 . . . . . . . 8  |-  F/_ x <. ( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `  y
) >.
23 fveq2 5695 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  A ) `  x
)  =  ( ( x  e.  Z  |->  A ) `  y ) )
24 fveq2 5695 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  B ) `  x
)  =  ( ( x  e.  Z  |->  B ) `  y ) )
2523, 24opeq12d 3960 . . . . . . . 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 4267 . . . . . . 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 17999 . . . . . 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 889 . . . . 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 448 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Z )  ->  x  e.  Z )
3015fvmpt2 5779 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  A  e.  X )  ->  ( ( x  e.  Z  |->  A ) `  x )  =  A )
3129, 7, 30syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  A ) `  x
)  =  A )
3217fvmpt2 5779 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  B  e.  Y )  ->  ( ( x  e.  Z  |->  B ) `  x )  =  B )
3329, 8, 32syl2anc 643 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  B ) `  x
)  =  B )
3431, 33opeq12d 3960 . . . . . . 7  |-  ( (
ph  /\  x  e.  Z )  ->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>.  =  <. A ,  B >. )
3534mpteq2dva 4263 . . . . . 6  |-  ( ph  ->  ( x  e.  Z  |-> 
<. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
3635fveq2d 5699 . . . . 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 2488 . . . 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 17997 . . . 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 1192 . . 3  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) ) )
41 eqidd 2413 . . . . 5  |-  ( ph  ->  ( x  e.  Z  |-> 
<. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
42 cnptop2 17269 . . . . . . . . 9  |-  ( O  e.  ( ( ( J  tX  K )  CnP  N ) `  <. R ,  S >. )  ->  N  e.  Top )
4338, 42syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
44 eqid 2412 . . . . . . . . 9  |-  U. N  =  U. N
4544toptopon 16961 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
4643, 45sylib 189 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
47 cnpf2 17276 . . . . . . 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 1184 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
4948feqmptd 5746 . . . . 5  |-  ( ph  ->  O  =  ( y  e.  ( X  X.  Y )  |->  ( O `
 y ) ) )
50 fveq2 5695 . . . . . 6  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( O `  <. A ,  B >. ) )
51 df-ov 6051 . . . . . 6  |-  ( A O B )  =  ( O `  <. A ,  B >. )
5250, 51syl6eqr 2462 . . . . 5  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( A O B ) )
5310, 41, 49, 52fmptco 5868 . . . 4  |-  ( ph  ->  ( O  o.  (
x  e.  Z  |->  <. A ,  B >. ) )  =  ( x  e.  Z  |->  ( A O B ) ) )
5453fveq2d 5699 . . 3  |-  ( ph  ->  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) )  =  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
5540, 54eleqtrd 2488 . 2  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
561, 55syl5eqel 2496 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 359    = wceq 1649    e. wcel 1721   <.cop 3785   U.cuni 3983    e. cmpt 4234    X. cxp 4843    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048   Topctop 16921  TopOnctopon 16922    CnP ccnp 17251    tX ctx 17553   Filcfil 17838    fLimf cflf 17928
This theorem is referenced by:  tsmsadd  18137
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-map 6987  df-topgen 13630  df-fbas 16662  df-fg 16663  df-top 16926  df-bases 16928  df-topon 16929  df-ntr 17047  df-nei 17125  df-cnp 17254  df-tx 17555  df-fil 17839  df-fm 17931  df-flim 17932  df-flf 17933
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