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Theorem txtopon 19820
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 19187 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  R  e.  Top )
2 topontop 19187 . . 3  |-  ( S  e.  (TopOn `  Y
)  ->  S  e.  Top )
3 txtop 19798 . . 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 2460 . . . . 5  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
6 eqid 2460 . . . . 5  |-  U. R  =  U. R
7 eqid 2460 . . . . 5  |-  U. S  =  U. S
85, 6, 7txuni2 19794 . . . 4  |-  ( U. R  X.  U. S )  =  U. ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
9 toponuni 19188 . . . . 5  |-  ( R  e.  (TopOn `  X
)  ->  X  =  U. R )
10 toponuni 19188 . . . . 5  |-  ( S  e.  (TopOn `  Y
)  ->  Y  =  U. S )
11 xpeq12 5011 . . . . 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 19795 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e.  _V )
14 unitg 19228 . . . . 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 2527 . . 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 19793 . . . 4  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  =  (
topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
1817unieqd 4248 . . 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 2504 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( X  X.  Y )  =  U. ( R  tX  S ) )
20 istopon 19186 . 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 1374    e. wcel 1762   _Vcvv 3106   U.cuni 4238    X. cxp 4990   ran crn 4993   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   topGenctg 14682   Topctop 19154  TopOnctopon 19155    tX ctx 19789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-topgen 14688  df-top 19159  df-bases 19161  df-topon 19162  df-tx 19791
This theorem is referenced by:  txuni  19821  txcls  19833  tx1cn  19838  tx2cn  19839  txcnp  19849  txcnmpt  19853  txindis  19863  txdis1cn  19864  txlm  19877  lmcn2  19878  xkococn  19889  cnmpt12  19896  cnmpt2c  19899  cnmpt21  19900  cnmpt2t  19902  cnmpt22  19903  cnmpt22f  19904  cnmpt2res  19906  cnmptcom  19907  cnmpt2k  19917  ptunhmeo  20037  xpstopnlem1  20038  xkocnv  20043  xkohmeo  20044  txflf  20235  flfcnp2  20236  cnmpt2plusg  20315  tmdcn2  20316  indistgp  20327  clssubg  20335  divstgplem  20347  prdstmdd  20350  tsmsadd  20377  cnmpt2vsca  20425  txmetcn  20779  cnmpt2ds  21076  fsum2cn  21103  cnmpt2pc  21156  htpyco2  21207  phtpyco2  21218  cnmpt2ip  21416  limccnp2  22024  dvcnp2  22051  dvaddbr  22069  dvmulbr  22070  dvcobr  22077  lhop1lem  22142  taylthlem2  22496  cxpcn3  22843  tpr2tp  27372  txsconlem  28175  txscon  28176  cvmlift2lem11  28248  cvmlift2lem12  28249
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