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Theorem txcmpb 17676
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 448 . . . . 5  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( R  tX  S )  e. 
Comp )
2 simplrr 738 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  Y  =/=  (/) )
3 fo1stres 6370 . . . . . . 7  |-  ( Y  =/=  (/)  ->  ( 1st  |`  ( X  X.  Y
) ) : ( X  X.  Y )
-onto-> X )
42, 3syl 16 . . . . . 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 17624 . . . . . . . 8  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( X  X.  Y
)  =  U. ( R  tX  S ) )
87ad2antrr 707 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( X  X.  Y )  = 
U. ( R  tX  S ) )
9 foeq2 5650 . . . . . . 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 16 . . . . . 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 202 . . . . 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 16998 . . . . . . 7  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
136toptopon 16998 . . . . . . 7  |-  ( S  e.  Top  <->  S  e.  (TopOn `  Y ) )
14 tx1cn 17641 . . . . . . 7  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 1st  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  R ) )
1512, 13, 14syl2anb 466 . . . . . 6  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( 1st  |`  ( X  X.  Y ) )  e.  ( ( R 
tX  S )  Cn  R ) )
1615ad2antrr 707 . . . . 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 17455 . . . . 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 737 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  X  =/=  (/) )
20 fo2ndres 6371 . . . . . . 7  |-  ( X  =/=  (/)  ->  ( 2nd  |`  ( X  X.  Y
) ) : ( X  X.  Y )
-onto-> Y )
2119, 20syl 16 . . . . . 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 5650 . . . . . . 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 16 . . . . . 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 202 . . . . 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 17642 . . . . . . 7  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( 2nd  |`  ( X  X.  Y
) )  e.  ( ( R  tX  S
)  Cn  S ) )
2612, 13, 25syl2anb 466 . . . . . 6  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( 2nd  |`  ( X  X.  Y ) )  e.  ( ( R 
tX  S )  Cn  S ) )
2726ad2antrr 707 . . . . 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 17455 . . . . 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 519 . . 3  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( R  e.  Comp  /\  S  e.  Comp ) )
3130ex 424 . 2  |-  ( ( ( R  e.  Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  -> 
( ( R  tX  S )  e.  Comp  -> 
( R  e.  Comp  /\  S  e.  Comp )
) )
32 txcmp 17675 . 2  |-  ( ( R  e.  Comp  /\  S  e.  Comp )  ->  ( R  tX  S )  e. 
Comp )
3331, 32impbid1 195 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 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   U.cuni 4015    X. cxp 4876    |` cres 4880   -onto->wfo 5452   ` cfv 5454  (class class class)co 6081   1stc1st 6347   2ndc2nd 6348   Topctop 16958  TopOnctopon 16959    Cn ccn 17288   Compccmp 17449    tX ctx 17592
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-fin 7113  df-topgen 13667  df-top 16963  df-bases 16965  df-topon 16966  df-cn 17291  df-cmp 17450  df-tx 17594
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