MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txtopon Structured version   Visualization version   GIF version

Theorem txtopon 21304
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 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

Proof of Theorem txtopon
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 20641 . . 3 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2 topontop 20641 . . 3 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
3 txtop 21282 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 494 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2621 . . . . 5 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
6 eqid 2621 . . . . 5 𝑅 = 𝑅
7 eqid 2621 . . . . 5 𝑆 = 𝑆
85, 6, 7txuni2 21278 . . . 4 ( 𝑅 × 𝑆) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
9 toponuni 20642 . . . . 5 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
10 toponuni 20642 . . . . 5 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
11 xpeq12 5094 . . . . 5 ((𝑋 = 𝑅𝑌 = 𝑆) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
129, 10, 11syl2an 494 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
135txbasex 21279 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
14 unitg 20682 . . . . 5 (ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V → (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
1513, 14syl 17 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
168, 12, 153eqtr4a 2681 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
175txval 21277 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1817unieqd 4412 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1916, 18eqtr4d 2658 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
20 istopon 20640 . 2 ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = (𝑅 ×t 𝑆)))
214, 19, 20sylanbrc 697 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186   cuni 4402   × cxp 5072  ran crn 5075  cfv 5847  (class class class)co 6604  cmpt2 6606  topGenctg 16019  Topctop 20617  TopOnctopon 20618   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-tx 21275
This theorem is referenced by:  txuni  21305  txcls  21317  tx1cn  21322  tx2cn  21323  txcnp  21333  txcnmpt  21337  txindis  21347  txdis1cn  21348  txlm  21361  lmcn2  21362  xkococn  21373  cnmpt12  21380  cnmpt2c  21383  cnmpt21  21384  cnmpt2t  21386  cnmpt22  21387  cnmpt22f  21388  cnmpt2res  21390  cnmptcom  21391  cnmpt2k  21401  ptunhmeo  21521  xpstopnlem1  21522  xkocnv  21527  xkohmeo  21528  txflf  21720  flfcnp2  21721  cnmpt2plusg  21802  tmdcn2  21803  indistgp  21814  clssubg  21822  qustgplem  21834  prdstmdd  21837  tsmsadd  21860  cnmpt2vsca  21908  txmetcn  22263  cnmpt2ds  22554  fsum2cn  22582  cnmpt2pc  22635  htpyco2  22686  phtpyco2  22697  cnmpt2ip  22955  limccnp2  23562  dvcnp2  23589  dvaddbr  23607  dvmulbr  23608  dvcobr  23615  lhop1lem  23680  taylthlem2  24032  cxpcn3  24389  tpr2tp  29732  txsconnlem  30930  txsconn  30931  cvmlift2lem11  31003  cvmlift2lem12  31004
  Copyright terms: Public domain W3C validator