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Theorem txtopon 17606
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 16974 . . 3  |-  ( R  e.  (TopOn `  X
)  ->  R  e.  Top )
2 topontop 16974 . . 3  |-  ( S  e.  (TopOn `  Y
)  ->  S  e.  Top )
3 txtop 17584 . . 3  |-  ( ( R  e.  Top  /\  S  e.  Top )  ->  ( R  tX  S
)  e.  Top )
41, 2, 3syl2an 464 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  Top )
5 eqid 2430 . . . . 5  |-  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )  =  ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
6 eqid 2430 . . . . 5  |-  U. R  =  U. R
7 eqid 2430 . . . . 5  |-  U. S  =  U. S
85, 6, 7txuni2 17580 . . . 4  |-  ( U. R  X.  U. S )  =  U. ran  (
u  e.  R , 
v  e.  S  |->  ( u  X.  v ) )
9 toponuni 16975 . . . . 5  |-  ( R  e.  (TopOn `  X
)  ->  X  =  U. R )
10 toponuni 16975 . . . . 5  |-  ( S  e.  (TopOn `  Y
)  ->  Y  =  U. S )
11 xpeq12 4883 . . . . 5  |-  ( ( X  =  U. R  /\  Y  =  U. S )  ->  ( X  X.  Y )  =  ( U. R  X.  U. S ) )
129, 10, 11syl2an 464 . . . 4  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( X  X.  Y )  =  ( U. R  X.  U. S ) )
135txbasex 17581 . . . . 5  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) )  e.  _V )
14 unitg 17015 . . . . 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 2488 . . 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 17579 . . . 4  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  =  (
topGen `  ran  ( u  e.  R ,  v  e.  S  |->  ( u  X.  v ) ) ) )
1817unieqd 4013 . . 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 2465 . 2  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( X  X.  Y )  =  U. ( R  tX  S ) )
20 istopon 16973 . 2  |-  ( ( R  tX  S )  e.  (TopOn `  ( X  X.  Y ) )  <-> 
( ( R  tX  S )  e.  Top  /\  ( X  X.  Y
)  =  U. ( R  tX  S ) ) )
214, 19, 20sylanbrc 646 1  |-  ( ( R  e.  (TopOn `  X )  /\  S  e.  (TopOn `  Y )
)  ->  ( R  tX  S )  e.  (TopOn `  ( X  X.  Y
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2943   U.cuni 4002    X. cxp 4862   ran crn 4865   ` cfv 5440  (class class class)co 6067    e. cmpt2 6069   topGenctg 13648   Topctop 16941  TopOnctopon 16942    tX ctx 17575
This theorem is referenced by:  txuni  17607  txcls  17619  tx1cn  17624  tx2cn  17625  txcnp  17635  txcnmpt  17639  txindis  17649  txdis1cn  17650  txlm  17663  lmcn2  17664  xkococn  17675  cnmpt12  17682  cnmpt2c  17685  cnmpt21  17686  cnmpt2t  17688  cnmpt22  17689  cnmpt22f  17690  cnmpt2res  17692  cnmptcom  17693  cnmpt2k  17703  ptunhmeo  17823  xpstopnlem1  17824  xkocnv  17829  xkohmeo  17830  txflf  18021  flfcnp2  18022  cnmpt2plusg  18101  tmdcn2  18102  indistgp  18113  clssubg  18121  divstgplem  18133  prdstmdd  18136  tsmsadd  18159  cnmpt2vsca  18207  txmetcn  18561  cnmpt2ds  18857  fsum2cn  18884  cnmpt2pc  18936  htpyco2  18987  phtpyco2  18998  cnmpt2ip  19185  limccnp2  19762  dvcnp2  19789  dvaddbr  19807  dvmulbr  19808  dvcobr  19815  lhop1lem  19880  taylthlem2  20273  cxpcn3  20615  tpr2tp  24285  txsconlem  24910  txscon  24911  cvmlift2lem11  24983  cvmlift2lem12  24984
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-iun 4082  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-1st 6335  df-2nd 6336  df-topgen 13650  df-top 16946  df-bases 16948  df-topon 16949  df-tx 17577
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