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Theorem txcmpb 17334
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 449 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( R  tX  S )  e. 
Comp )
2 simplrr 739 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  Y  =/=  (/) )
3 fo1stres 6106 . . . . . . 7  |-  ( Y  =/=  (/)  ->  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y )
-onto-> X )
42, 3syl 17 . . . . . 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 17283 . . . . . . . 8  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( X  X.  Y
)  =  U. ( R  tX  S ) )
87ad2antrr 708 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( X  X.  Y )  = 
U. ( R  tX  S ) )
9 foeq2 5415 . . . . . . 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 17 . . . . . 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 203 . . . . 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 16667 . . . . . . 7  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
136toptopon 16667 . . . . . . 7  |-  ( S  e.  Top  <->  S  e.  (TopOn `  Y ) )
14 tx1cn 17299 . . . . . . 7  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  R ) )
1512, 13, 14syl2anb 467 . . . . . 6  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R 
tX  S )  Cn  R ) )
1615ad2antrr 708 . . . . 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 17115 . . . . 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 1184 . . . 4  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  R  e.  Comp )
19 simplrl 738 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  X  =/=  (/) )
20 fo2ndres 6107 . . . . . . 7  |-  ( X  =/=  (/)  ->  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y )
-onto-> Y )
2119, 20syl 17 . . . . . 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 5415 . . . . . . 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 17 . . . . . 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 203 . . . . 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 17300 . . . . . . 7  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  S ) )
2612, 13, 25syl2anb 467 . . . . . 6  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R 
tX  S )  Cn  S ) )
2726ad2antrr 708 . . . . 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 17115 . . . . 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 1184 . . . 4  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  S  e.  Comp )
3018, 29jca 520 . . 3  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( R  e.  Comp  /\  S  e.  Comp ) )
3130ex 425 . 2  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( ( R  tX  S )  e.  Comp  -> 
( R  e.  Comp  /\  S  e.  Comp )
) )
32 txcmp 17333 . 2  |-  ( ( R  e.  Comp  /\  S  e.  Comp )  ->  ( R  tX  S )  e. 
Comp )
3331, 32impbid1 196 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 6    <-> wb 178    /\ wa 360    = wceq 1625    e. wcel 1687    =/= wne 2449   (/)c0 3458   U.cuni 3830    X. cxp 4688    |` cres 4692   -onto->wfo 5221   ` cfv 5223  (class class class)co 5821   1stc1st 6083   2ndc2nd 6084   Topctop 16627  TopOnctopon 16628    Cn ccn 16950   Compccmp 17109    tX ctx 17251
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1638  ax-8 1646  ax-13 1689  ax-14 1691  ax-6 1706  ax-7 1711  ax-11 1718  ax-12 1870  ax-ext 2267  ax-rep 4134  ax-sep 4144  ax-nul 4152  ax-pow 4189  ax-pr 4215  ax-un 4513
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1531  df-nf 1534  df-sb 1633  df-eu 2150  df-mo 2151  df-clab 2273  df-cleq 2279  df-clel 2282  df-nfc 2411  df-ne 2451  df-ral 2551  df-rex 2552  df-reu 2553  df-rab 2555  df-v 2793  df-sbc 2995  df-csb 3085  df-dif 3158  df-un 3160  df-in 3162  df-ss 3169  df-pss 3171  df-nul 3459  df-if 3569  df-pw 3630  df-sn 3649  df-pr 3650  df-tp 3651  df-op 3652  df-uni 3831  df-int 3866  df-iun 3910  df-iin 3911  df-br 4027  df-opab 4081  df-mpt 4082  df-tr 4117  df-eprel 4306  df-id 4310  df-po 4315  df-so 4316  df-fr 4353  df-we 4355  df-ord 4396  df-on 4397  df-lim 4398  df-suc 4399  df-om 4658  df-xp 4696  df-rel 4697  df-cnv 4698  df-co 4699  df-dm 4700  df-rn 4701  df-res 4702  df-ima 4703  df-fun 5225  df-fn 5226  df-f 5227  df-f1 5228  df-fo 5229  df-f1o 5230  df-fv 5231  df-ov 5824  df-oprab 5825  df-mpt2 5826  df-1st 6085  df-2nd 6086  df-recs 6385  df-rdg 6420  df-1o 6476  df-oadd 6480  df-er 6657  df-map 6771  df-en 6861  df-dom 6862  df-fin 6864  df-topgen 13340  df-top 16632  df-bases 16634  df-topon 16635  df-cn 16953  df-cmp 17110  df-tx 17253
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