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Theorem txcmpb 17338
Description: The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
txcmpb.1  |-  X  = 
U. R
txcmpb.2  |-  Y  = 
U. S
Assertion
Ref Expression
txcmpb  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( ( R  tX  S )  e.  Comp  <->  ( R  e.  Comp  /\  S  e.  Comp ) ) )

Proof of Theorem txcmpb
StepHypRef Expression
1 simpr 447 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( R  tX  S )  e. 
Comp )
2 simplrr 737 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  Y  =/=  (/) )
3 fo1stres 6143 . . . . . . 7  |-  ( Y  =/=  (/)  ->  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y )
-onto-> X )
42, 3syl 15 . . . . . 6  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y
) -onto-> X )
5 txcmpb.1 . . . . . . . . 9  |-  X  = 
U. R
6 txcmpb.2 . . . . . . . . 9  |-  Y  = 
U. S
75, 6txuni 17287 . . . . . . . 8  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( X  X.  Y
)  =  U. ( R  tX  S ) )
87ad2antrr 706 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( X  X.  Y )  = 
U. ( R  tX  S ) )
9 foeq2 5448 . . . . . . 7  |-  ( ( X  X.  Y )  =  U. ( R 
tX  S )  -> 
( ( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> X  <->  ( 1st  |`  ( X  X.  Y
) ) : U. ( R  tX  S )
-onto-> X ) )
108, 9syl 15 . . . . . 6  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  (
( 1st  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> X  <->  ( 1st  |`  ( X  X.  Y
) ) : U. ( R  tX  S )
-onto-> X ) )
114, 10mpbid 201 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( 1st  |`  ( X  X.  Y ) ) : U. ( R  tX  S ) -onto-> X )
125toptopon 16671 . . . . . . 7  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
136toptopon 16671 . . . . . . 7  |-  ( S  e.  Top  <->  S  e.  (TopOn `  Y ) )
14 tx1cn 17303 . . . . . . 7  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  R ) )
1512, 13, 14syl2anb 465 . . . . . 6  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R 
tX  S )  Cn  R ) )
1615ad2antrr 706 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  R
) )
175cncmp 17119 . . . . 5  |-  ( ( ( R  tX  S
)  e.  Comp  /\  ( 1st  |`  ( X  X.  Y ) ) : U. ( R  tX  S ) -onto-> X  /\  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  R
) )  ->  R  e.  Comp )
181, 11, 16, 17syl3anc 1182 . . . 4  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  R  e.  Comp )
19 simplrl 736 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  X  =/=  (/) )
20 fo2ndres 6144 . . . . . . 7  |-  ( X  =/=  (/)  ->  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y )
-onto-> Y )
2119, 20syl 15 . . . . . 6  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y
) -onto-> Y )
22 foeq2 5448 . . . . . . 7  |-  ( ( X  X.  Y )  =  U. ( R 
tX  S )  -> 
( ( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Y  <->  ( 2nd  |`  ( X  X.  Y
) ) : U. ( R  tX  S )
-onto-> Y ) )
238, 22syl 15 . . . . . 6  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  (
( 2nd  |`  ( X  X.  Y ) ) : ( X  X.  Y ) -onto-> Y  <->  ( 2nd  |`  ( X  X.  Y
) ) : U. ( R  tX  S )
-onto-> Y ) )
2421, 23mpbid 201 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( 2nd  |`  ( X  X.  Y ) ) : U. ( R  tX  S ) -onto-> Y )
25 tx2cn 17304 . . . . . . 7  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  S ) )
2612, 13, 25syl2anb 465 . . . . . 6  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R 
tX  S )  Cn  S ) )
2726ad2antrr 706 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  S
) )
286cncmp 17119 . . . . 5  |-  ( ( ( R  tX  S
)  e.  Comp  /\  ( 2nd  |`  ( X  X.  Y ) ) : U. ( R  tX  S ) -onto-> Y  /\  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R  tX  S )  Cn  S
) )  ->  S  e.  Comp )
291, 24, 27, 28syl3anc 1182 . . . 4  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  S  e.  Comp )
3018, 29jca 518 . . 3  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( R  e.  Comp  /\  S  e.  Comp ) )
3130ex 423 . 2  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( ( R  tX  S )  e.  Comp  -> 
( R  e.  Comp  /\  S  e.  Comp )
) )
32 txcmp 17337 . 2  |-  ( ( R  e.  Comp  /\  S  e.  Comp )  ->  ( R  tX  S )  e. 
Comp )
3331, 32impbid1 194 1  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( ( R  tX  S )  e.  Comp  <->  ( R  e.  Comp  /\  S  e.  Comp ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   U.cuni 3827    X. cxp 4687    |` cres 4691   -onto->wfo 5253   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   Topctop 16631  TopOnctopon 16632    Cn ccn 16954   Compccmp 17113    tX ctx 17255
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-3or 935  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-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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-fin 6867  df-topgen 13344  df-top 16636  df-bases 16638  df-topon 16639  df-cn 16957  df-cmp 17114  df-tx 17257
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