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Theorem limptlimpr 25576
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 25574 . 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 25575 . . 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 1623    e. wcel 1684   <.cop 3643   U.cuni 3827    X. cxp 4687    o. ccom 4693   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Topctop 16631    tX ctx 17255   Filcfil 17540    fLimf cflf 17630
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-nei 16835  df-tx 17257  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635
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