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Theorem limptlimpr 25679
Description: A limit in a product topology exists iff the limits of the projections exist. (Contributed by FL, 15-Oct-2012.) (Revised by Mario Carneiro, 9-Aug-2015.)
Hypotheses
Ref Expression
limptlimpr.t  |-  T  =  ( R  tX  S
)
limptlimpr.x  |-  X  = 
U. R
limptlimpr.y  |-  Y  = 
U. S
limptlimpr.z  |-  Z  =  ( X  X.  Y
)
Assertion
Ref Expression
limptlimpr  |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A
) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T 
fLimf  L ) `  F
)  <->  ( L1  e.  ( ( R  fLimf  L ) `  ( 1st 
o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd 
o.  F ) ) ) ) )

Proof of Theorem limptlimpr
StepHypRef Expression
1 limptlimpr.t . . 3  |-  T  =  ( R  tX  S
)
2 limptlimpr.x . . 3  |-  X  = 
U. R
3 limptlimpr.y . . 3  |-  Y  = 
U. S
4 limptlimpr.z . . 3  |-  Z  =  ( X  X.  Y
)
51, 2, 3, 4limptlimpr2lem1 25677 . 2  |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A
) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T 
fLimf  L ) `  F
)  ->  ( L1  e.  ( ( R  fLimf  L ) `  ( 1st 
o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd 
o.  F ) ) ) ) )
61, 2, 3, 4limptlimpr2lem2 25678 . . 3  |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  F : W --> Z )  ->  ( ( L1  e.  ( ( R  fLimf  L ) `  ( 1st 
o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd 
o.  F ) ) )  ->  <. L1 ,  L 2 >.  e.  (
( T  fLimf  L ) `
 F ) ) )
76adantrr 697 . 2  |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A
) )  ->  (
( L1  e.  (
( R  fLimf  L ) `
 ( 1st  o.  F ) )  /\  L 2  e.  (
( S  fLimf  L ) `
 ( 2nd  o.  F ) ) )  ->  <. L1 ,  L 2 >.  e.  ( ( T 
fLimf  L ) `  F
) ) )
85, 7impbid 183 1  |-  ( ( ( R  e.  Top  /\  S  e.  Top  /\  L  e.  ( Fil `  W ) )  /\  ( F : W --> Z  /\  L 2  e.  A
) )  ->  ( <. L1 ,  L 2 >.  e.  ( ( T 
fLimf  L ) `  F
)  <->  ( L1  e.  ( ( R  fLimf  L ) `  ( 1st 
o.  F ) )  /\  L 2  e.  ( ( S  fLimf  L ) `  ( 2nd 
o.  F ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   U.cuni 3843    X. cxp 4703    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Topctop 16647    tX ctx 17271   Filcfil 17556    fLimf cflf 17646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-nei 16851  df-tx 17273  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651
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