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Theorem txcmpb 17300
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 740 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  Y  =/=  (/) )
3 fo1stres 6077 . . . . . . 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 17249 . . . . . . . 8  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( X  X.  Y
)  =  U. ( R  tX  S ) )
87ad2antrr 709 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  ( X  X.  Y )  = 
U. ( R  tX  S ) )
9 foeq2 5386 . . . . . . 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 16633 . . . . . . 7  |-  ( R  e.  Top  <->  R  e.  (TopOn `  X ) )
136toptopon 16633 . . . . . . 7  |-  ( S  e.  Top  <->  S  e.  (TopOn `  Y ) )
14 tx1cn 17265 . . . . . . 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 709 . . . . 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 17081 . . . . 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 1187 . . . 4  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  R  e.  Comp )
19 simplrl 739 . . . . . . 7  |-  ( ( ( ( R  e. 
Top  /\  S  e.  Top )  /\  ( X  =/=  (/)  /\  Y  =/=  (/) ) )  /\  ( R  tX  S )  e. 
Comp )  ->  X  =/=  (/) )
20 fo2ndres 6078 . . . . . . 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 5386 . . . . . . 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 17266 . . . . . . 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 709 . . . . 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 17081 . . . . 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 1187 . . . 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 17299 . 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 1619    e. wcel 1621    =/= wne 2421   (/)c0 3430   U.cuni 3801    X. cxp 4659    |` cres 4663   -onto->wfo 4671   ` cfv 4673  (class class class)co 5792   1stc1st 6054   2ndc2nd 6055   Topctop 16593  TopOnctopon 16594    Cn ccn 16916   Compccmp 17075    tX ctx 17217
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-1st 6056  df-2nd 6057  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-er 6628  df-map 6742  df-en 6832  df-dom 6833  df-fin 6835  df-topgen 13306  df-top 16598  df-bases 16600  df-topon 16601  df-cn 16919  df-cmp 17076  df-tx 17219
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