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Theorem txtopon 19280
Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
txtopon  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )

Proof of Theorem txtopon
Dummy variables  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 18647 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  R  e.  Top )
2 topontop 18647 . . 3  |-  ( S  e.  (TopOn `  Y
)  ->  S  e.  Top )
3 txtop 19258 . . 3  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( R  tX  S
)  e.  Top )
41, 2, 3syl2an 477 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  Top )
5 eqid 2451 . . . . 5  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
6 eqid 2451 . . . . 5  |-  U. R  =  U. R
7 eqid 2451 . . . . 5  |-  U. S  =  U. S
85, 6, 7txuni2 19254 . . . 4  |-  ( U. R  X.  U. S )  =  U. ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
9 toponuni 18648 . . . . 5  |-  ( R  e.  (TopOn `  X
)  ->  X  =  U. R )
10 toponuni 18648 . . . . 5  |-  ( S  e.  (TopOn `  Y
)  ->  Y  =  U. S )
11 xpeq12 4957 . . . . 5  |-  ( ( X  =  U. R  /\  Y  =  U. S )  ->  ( X  X.  Y )  =  ( U. R  X.  U. S ) )
129, 10, 11syl2an 477 . . . 4  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( X  X.  Y )  =  ( U. R  X.  U. S ) )
135txbasex 19255 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e.  _V )
14 unitg 18688 . . . . 5  |-  ( ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) )  e.  _V  ->  U. ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) ) )  = 
U. ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) )
1513, 14syl 16 . . . 4  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  U. ( topGen `
 ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) )  =  U. ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) ) )
168, 12, 153eqtr4a 2518 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( X  X.  Y )  =  U. ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
175txval 19253 . . . 4  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  =  (
topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
1817unieqd 4199 . . 3  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  U. ( R  tX  S )  = 
U. ( topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v
) ) ) )
1916, 18eqtr4d 2495 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( X  X.  Y )  =  U. ( R  tX  S ) )
20 istopon 18646 . 2  |-  ( ( R  tX  S )  e.  (TopOn `  ( X  X.  Y ) )  <-> 
( ( R  tX  S )  e.  Top  /\  ( X  X.  Y
)  =  U. ( R  tX  S ) ) )
214, 19, 20sylanbrc 664 1  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3068   U.cuni 4189    X. cxp 4936   ran crn 4939   ` cfv 5516  (class class class)co 6190    |-> cmpt2 6192   topGenctg 14478   Topctop 18614  TopOnctopon 18615    tX ctx 19249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fv 5524  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-1st 6677  df-2nd 6678  df-topgen 14484  df-top 18619  df-bases 18621  df-topon 18622  df-tx 19251
This theorem is referenced by:  txuni  19281  txcls  19293  tx1cn  19298  tx2cn  19299  txcnp  19309  txcnmpt  19313  txindis  19323  txdis1cn  19324  txlm  19337  lmcn2  19338  xkococn  19349  cnmpt12  19356  cnmpt2c  19359  cnmpt21  19360  cnmpt2t  19362  cnmpt22  19363  cnmpt22f  19364  cnmpt2res  19366  cnmptcom  19367  cnmpt2k  19377  ptunhmeo  19497  xpstopnlem1  19498  xkocnv  19503  xkohmeo  19504  txflf  19695  flfcnp2  19696  cnmpt2plusg  19775  tmdcn2  19776  indistgp  19787  clssubg  19795  divstgplem  19807  prdstmdd  19810  tsmsadd  19837  cnmpt2vsca  19885  txmetcn  20239  cnmpt2ds  20536  fsum2cn  20563  cnmpt2pc  20616  htpyco2  20667  phtpyco2  20678  cnmpt2ip  20876  limccnp2  21483  dvcnp2  21510  dvaddbr  21528  dvmulbr  21529  dvcobr  21536  lhop1lem  21601  taylthlem2  21955  cxpcn3  22302  tpr2tp  26468  txsconlem  27263  txscon  27264  cvmlift2lem11  27336  cvmlift2lem12  27337
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