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Theorem lmcn2 17683
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 5872 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  X )
3 txlm.g . . . . . . 7  |-  ( ph  ->  G : Z --> Y )
43ffvelrnda 5872 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( G `  n )  e.  Y )
5 opelxpi 4912 . . . . . 6  |-  ( ( ( F `  n
)  e.  X  /\  ( G `  n )  e.  Y )  ->  <. ( F `  n
) ,  ( G `
 n ) >.  e.  ( X  X.  Y
) )
62, 4, 5syl2anc 644 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  <. ( F `  n ) ,  ( G `  n ) >.  e.  ( X  X.  Y ) )
7 eqidd 2439 . . . . 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 17625 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
118, 9, 10syl2anc 644 . . . . . . 7  |-  ( ph  ->  ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) ) )
12 lmcn2.o . . . . . . . . 9  |-  ( ph  ->  O  e.  ( ( J  tX  K )  Cn  N ) )
13 cntop2 17307 . . . . . . . . 9  |-  ( O  e.  ( ( J 
tX  K )  Cn  N )  ->  N  e.  Top )
1412, 13syl 16 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
15 eqid 2438 . . . . . . . . 9  |-  U. N  =  U. N
1615toptopon 17000 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1714, 16sylib 190 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
18 cnf2 17315 . . . . . . 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 1185 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
2019feqmptd 5781 . . . . 5  |-  ( ph  ->  O  =  ( x  e.  ( X  X.  Y )  |->  ( O `
 x ) ) )
21 fveq2 5730 . . . . . 6  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
)
22 df-ov 6086 . . . . . 6  |-  ( ( F `  n ) O ( G `  n ) )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
2321, 22syl6eqr 2488 . . . . 5  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( ( F `  n
) O ( G `
 n ) ) )
246, 7, 20, 23fmptco 5903 . . . 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 2488 . . 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 2438 . . . . . 6  |-  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n )
>. )
3229, 30, 8, 9, 1, 3, 31txlm 17682 . . . . 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 889 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
<. ( F `  n
) ,  ( G `
 n ) >.
) ( ~~> t `  ( J  tX  K ) ) <. R ,  S >. )
3433, 12lmcn 17371 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) ) ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
3526, 34eqbrtrrd 4236 . 2  |-  ( ph  ->  H ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
36 df-ov 6086 . 2  |-  ( R O S )  =  ( O `  <. R ,  S >. )
3735, 36syl6breqr 4254 1  |-  ( ph  ->  H ( ~~> t `  N ) ( R O S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   <.cop 3819   U.cuni 4017   class class class wbr 4214    e. cmpt 4268    X. cxp 4878    o. ccom 4884   -->wf 5452   ` cfv 5456  (class class class)co 6083   ZZcz 10284   ZZ>=cuz 10490   Topctop 16960  TopOnctopon 16961    Cn ccn 17290   ~~> tclm 17292    tX ctx 17594
This theorem is referenced by:  hlimadd  22697
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  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-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-map 7022  df-pm 7023  df-en 7112  df-dom 7113  df-sdom 7114  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-nn 10003  df-z 10285  df-uz 10491  df-topgen 13669  df-top 16965  df-bases 16967  df-topon 16968  df-cn 17293  df-cnp 17294  df-lm 17295  df-tx 17596
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