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Theorem tpr2tp 27510
Description: The usual topology on  ( RR  X.  RR ) is the product topology of the usual topology on  RR. (Contributed by Thierry Arnoux, 21-Sep-2017.)
Hypothesis
Ref Expression
tpr2tp.0  |-  J  =  ( topGen `  ran  (,) )
Assertion
Ref Expression
tpr2tp  |-  ( J 
tX  J )  e.  (TopOn `  ( RR  X.  RR ) )

Proof of Theorem tpr2tp
StepHypRef Expression
1 tpr2tp.0 . . 3  |-  J  =  ( topGen `  ran  (,) )
2 retopon 21000 . . 3  |-  ( topGen ` 
ran  (,) )  e.  (TopOn `  RR )
31, 2eqeltri 2546 . 2  |-  J  e.  (TopOn `  RR )
4 txtopon 19822 . 2  |-  ( ( J  e.  (TopOn `  RR )  /\  J  e.  (TopOn `  RR )
)  ->  ( J  tX  J )  e.  (TopOn `  ( RR  X.  RR ) ) )
53, 3, 4mp2an 672 1  |-  ( J 
tX  J )  e.  (TopOn `  ( RR  X.  RR ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374    e. wcel 1762    X. cxp 4992   ran crn 4995   ` cfv 5581  (class class class)co 6277   RRcr 9482   (,)cioo 11520   topGenctg 14684  TopOnctopon 19157    tX ctx 19791
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-pre-lttri 9557  ax-pre-lttrn 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-po 4795  df-so 4796  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6776  df-2nd 6777  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-ioo 11524  df-topgen 14690  df-top 19161  df-bases 19163  df-topon 19164  df-tx 19793
This theorem is referenced by:  tpr2uni  27511  sxbrsigalem4  27886  sxbrsiga  27889
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