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Theorem txcmpb 17354
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 6159 . . . . . . 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 17303 . . . . . . . 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 5464 . . . . . . 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 16687 . . . . . . 7  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
136toptopon 16687 . . . . . . 7  |-  ( S  e.  Top  <->  S  e.  (TopOn `  Y ) )
14 tx1cn 17319 . . . . . . 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 17135 . . . . 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 6160 . . . . . . 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 5464 . . . . . . 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 17320 . . . . . . 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 17135 . . . . 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 17353 . 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 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   U.cuni 3843    X. cxp 4703    |` cres 4707   -onto->wfo 5269   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137   Topctop 16647  TopOnctopon 16648    Cn ccn 16970   Compccmp 17129    tX ctx 17271
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-3or 935  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-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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-fin 6883  df-topgen 13360  df-top 16652  df-bases 16654  df-topon 16655  df-cn 16973  df-cmp 17130  df-tx 17273
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