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Theorem lmcn2 20440
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 6008 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( F `  n )  e.  X )
3 txlm.g . . . . . . 7  |-  ( ph  ->  G : Z --> Y )
43ffvelrnda 6008 . . . . . 6  |-  ( (
ph  /\  n  e.  Z )  ->  ( G `  n )  e.  Y )
5 opelxpi 4854 . . . . . 6  |-  ( ( ( F `  n
)  e.  X  /\  ( G `  n )  e.  Y )  ->  <. ( F `  n
) ,  ( G `
 n ) >.  e.  ( X  X.  Y
) )
62, 4, 5syl2anc 659 . . . . 5  |-  ( (
ph  /\  n  e.  Z )  ->  <. ( F `  n ) ,  ( G `  n ) >.  e.  ( X  X.  Y ) )
7 eqidd 2403 . . . . 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 20382 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  K  e.  (TopOn `  Y )
)  ->  ( J  tX  K )  e.  (TopOn `  ( X  X.  Y
) ) )
118, 9, 10syl2anc 659 . . . . . . 7  |-  ( ph  ->  ( J  tX  K
)  e.  (TopOn `  ( X  X.  Y
) ) )
12 lmcn2.o . . . . . . . . 9  |-  ( ph  ->  O  e.  ( ( J  tX  K )  Cn  N ) )
13 cntop2 20033 . . . . . . . . 9  |-  ( O  e.  ( ( J 
tX  K )  Cn  N )  ->  N  e.  Top )
1412, 13syl 17 . . . . . . . 8  |-  ( ph  ->  N  e.  Top )
15 eqid 2402 . . . . . . . . 9  |-  U. N  =  U. N
1615toptopon 19724 . . . . . . . 8  |-  ( N  e.  Top  <->  N  e.  (TopOn `  U. N ) )
1714, 16sylib 196 . . . . . . 7  |-  ( ph  ->  N  e.  (TopOn `  U. N ) )
18 cnf2 20041 . . . . . . 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 1230 . . . . . 6  |-  ( ph  ->  O : ( X  X.  Y ) --> U. N )
2019feqmptd 5901 . . . . 5  |-  ( ph  ->  O  =  ( x  e.  ( X  X.  Y )  |->  ( O `
 x ) ) )
21 fveq2 5848 . . . . . 6  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
)
22 df-ov 6280 . . . . . 6  |-  ( ( F `  n ) O ( G `  n ) )  =  ( O `  <. ( F `  n ) ,  ( G `  n ) >. )
2321, 22syl6eqr 2461 . . . . 5  |-  ( x  =  <. ( F `  n ) ,  ( G `  n )
>.  ->  ( O `  x )  =  ( ( F `  n
) O ( G `
 n ) ) )
246, 7, 20, 23fmptco 6042 . . . 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 2461 . . 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 2402 . . . . . 6  |-  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n ) >. )  =  ( n  e.  Z  |->  <. ( F `  n ) ,  ( G `  n )
>. )
3229, 30, 8, 9, 1, 3, 31txlm 20439 . . . . 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 922 . . . 4  |-  ( ph  ->  ( n  e.  Z  |-> 
<. ( F `  n
) ,  ( G `
 n ) >.
) ( ~~> t `  ( J  tX  K ) ) <. R ,  S >. )
3433, 12lmcn 20097 . . 3  |-  ( ph  ->  ( O  o.  (
n  e.  Z  |->  <.
( F `  n
) ,  ( G `
 n ) >.
) ) ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
3526, 34eqbrtrrd 4416 . 2  |-  ( ph  ->  H ( ~~> t `  N ) ( O `
 <. R ,  S >. ) )
36 df-ov 6280 . 2  |-  ( R O S )  =  ( O `  <. R ,  S >. )
3735, 36syl6breqr 4434 1  |-  ( ph  ->  H ( ~~> t `  N ) ( R O S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842   <.cop 3977   U.cuni 4190   class class class wbr 4394    |-> cmpt 4452    X. cxp 4820    o. ccom 4826   -->wf 5564   ` cfv 5568  (class class class)co 6277   ZZcz 10904   ZZ>=cuz 11126   Topctop 19684  TopOnctopon 19685    Cn ccn 20016   ~~> tclm 20018    tX ctx 20351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-z 10905  df-uz 11127  df-topgen 15056  df-top 19689  df-bases 19691  df-topon 19692  df-cn 20019  df-cnp 20020  df-lm 20021  df-tx 20353
This theorem is referenced by:  hlimadd  26510
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