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Theorem flfcnp2 18044
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 6087 . 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 17628 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
52, 3, 4syl2anc 644 . . . 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 4913 . . . . . 6  |-  ( ( A  e.  X  /\  B  e.  Y )  -> 
<. A ,  B >.  e.  ( X  X.  Y
) )
107, 8, 9syl2anc 644 . . . . 5  |-  ( (
ph  /\  x  e.  Z )  ->  <. A ,  B >.  e.  ( X  X.  Y ) )
11 eqid 2438 . . . . 5  |-  ( x  e.  Z  |->  <. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. )
1210, 11fmptd 5896 . . . 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 2438 . . . . . . . 8  |-  ( x  e.  Z  |->  A )  =  ( x  e.  Z  |->  A )
167, 15fmptd 5896 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  A ) : Z --> X )
17 eqid 2438 . . . . . . . 8  |-  ( x  e.  Z  |->  B )  =  ( x  e.  Z  |->  B )
188, 17fmptd 5896 . . . . . . 7  |-  ( ph  ->  ( x  e.  Z  |->  B ) : Z --> Y )
19 nfcv 2574 . . . . . . . 8  |-  F/_ y <. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >.
20 nffvmpt1 5739 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  A ) `  y )
21 nffvmpt1 5739 . . . . . . . . 9  |-  F/_ x
( ( x  e.  Z  |->  B ) `  y )
2220, 21nfop 4002 . . . . . . . 8  |-  F/_ x <. ( ( x  e.  Z  |->  A ) `  y ) ,  ( ( x  e.  Z  |->  B ) `  y
) >.
23 fveq2 5731 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  A ) `  x
)  =  ( ( x  e.  Z  |->  A ) `  y ) )
24 fveq2 5731 . . . . . . . . 9  |-  ( x  =  y  ->  (
( x  e.  Z  |->  B ) `  x
)  =  ( ( x  e.  Z  |->  B ) `  y ) )
2523, 24opeq12d 3994 . . . . . . . 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 4302 . . . . . . 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 18043 . . . . . 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 890 . . . . 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 449 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  Z )  ->  x  e.  Z )
3015fvmpt2 5815 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  A  e.  X )  ->  ( ( x  e.  Z  |->  A ) `  x )  =  A )
3129, 7, 30syl2anc 644 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  A ) `  x
)  =  A )
3217fvmpt2 5815 . . . . . . . . 9  |-  ( ( x  e.  Z  /\  B  e.  Y )  ->  ( ( x  e.  Z  |->  B ) `  x )  =  B )
3329, 8, 32syl2anc 644 . . . . . . . 8  |-  ( (
ph  /\  x  e.  Z )  ->  (
( x  e.  Z  |->  B ) `  x
)  =  B )
3431, 33opeq12d 3994 . . . . . . 7  |-  ( (
ph  /\  x  e.  Z )  ->  <. (
( x  e.  Z  |->  A ) `  x
) ,  ( ( x  e.  Z  |->  B ) `  x )
>.  =  <. A ,  B >. )
3534mpteq2dva 4298 . . . . . 6  |-  ( ph  ->  ( x  e.  Z  |-> 
<. ( ( x  e.  Z  |->  A ) `  x ) ,  ( ( x  e.  Z  |->  B ) `  x
) >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
3635fveq2d 5735 . . . . 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 2514 . . . 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 18041 . . . 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 1193 . . 3  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) ) )
41 eqidd 2439 . . . . 5  |-  ( ph  ->  ( x  e.  Z  |-> 
<. A ,  B >. )  =  ( x  e.  Z  |->  <. A ,  B >. ) )
42 cnptop2 17312 . . . . . . . . 9  |-  ( O  e.  ( ( ( J  tX  K )  CnP  N ) `  <. R ,  S >. )  ->  N  e.  Top )
4338, 42syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
44 eqid 2438 . . . . . . . . 9  |-  U. N  =  U. N
4544toptopon 17003 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
4643, 45sylib 190 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
47 cnpf2 17319 . . . . . . 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 1185 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
4948feqmptd 5782 . . . . 5  |-  ( ph  ->  O  =  ( y  e.  ( X  X.  Y )  |->  ( O `
 y ) ) )
50 fveq2 5731 . . . . . 6  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( O `  <. A ,  B >. ) )
51 df-ov 6087 . . . . . 6  |-  ( A O B )  =  ( O `  <. A ,  B >. )
5250, 51syl6eqr 2488 . . . . 5  |-  ( y  =  <. A ,  B >.  ->  ( O `  y )  =  ( A O B ) )
5310, 41, 49, 52fmptco 5904 . . . 4  |-  ( ph  ->  ( O  o.  (
x  e.  Z  |->  <. A ,  B >. ) )  =  ( x  e.  Z  |->  ( A O B ) ) )
5453fveq2d 5735 . . 3  |-  ( ph  ->  ( ( N  fLimf  L ) `  ( O  o.  ( x  e.  Z  |->  <. A ,  B >. ) ) )  =  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
5540, 54eleqtrd 2514 . 2  |-  ( ph  ->  ( O `  <. R ,  S >. )  e.  ( ( N  fLimf  L ) `  ( x  e.  Z  |->  ( A O B ) ) ) )
561, 55syl5eqel 2522 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 360    = wceq 1653    e. wcel 1726   <.cop 3819   U.cuni 4017    e. cmpt 4269    X. cxp 4879    o. ccom 4885   -->wf 5453   ` cfv 5457  (class class class)co 6084   Topctop 16963  TopOnctopon 16964    CnP ccnp 17294    tX ctx 17597   Filcfil 17882    fLimf cflf 17972
This theorem is referenced by:  tsmsadd  18181
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-map 7023  df-topgen 13672  df-fbas 16704  df-fg 16705  df-top 16968  df-bases 16970  df-topon 16971  df-ntr 17089  df-nei 17167  df-cnp 17297  df-tx 17599  df-fil 17883  df-fm 17975  df-flim 17976  df-flf 17977
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