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Theorem txtopon 22127
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 21449 . . 3 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2 topontop 21449 . . 3 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
3 txtop 22105 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 595 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2818 . . . . 5 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
6 eqid 2818 . . . . 5 𝑅 = 𝑅
7 eqid 2818 . . . . 5 𝑆 = 𝑆
85, 6, 7txuni2 22101 . . . 4 ( 𝑅 × 𝑆) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
9 toponuni 21450 . . . . 5 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
10 toponuni 21450 . . . . 5 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
11 xpeq12 5573 . . . . 5 ((𝑋 = 𝑅𝑌 = 𝑆) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
129, 10, 11syl2an 595 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
135txbasex 22102 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
14 unitg 21503 . . . . 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 2879 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
175txval 22100 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1817unieqd 4840 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1916, 18eqtr4d 2856 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
20 istopon 21448 . 2 ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = (𝑅 ×t 𝑆)))
214, 19, 20sylanbrc 583 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492   cuni 4830   × cxp 5546  ran crn 5549  cfv 6348  (class class class)co 7145  cmpo 7147  topGenctg 16699  Topctop 21429  TopOnctopon 21446   ×t ctx 22096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-topgen 16705  df-top 21430  df-topon 21447  df-bases 21482  df-tx 22098
This theorem is referenced by:  txuni  22128  txcls  22140  tx1cn  22145  tx2cn  22146  txcnp  22156  txcnmpt  22160  txindis  22170  txdis1cn  22171  txlm  22184  lmcn2  22185  xkococn  22196  cnmpt12  22203  cnmpt2c  22206  cnmpt21  22207  cnmpt2t  22209  cnmpt22  22210  cnmpt22f  22211  cnmpt2res  22213  cnmptcom  22214  cnmpt2k  22224  ptunhmeo  22344  xpstopnlem1  22345  xkocnv  22350  xkohmeo  22351  txflf  22542  flfcnp2  22543  cnmpt2plusg  22624  tmdcn2  22625  indistgp  22636  clssubg  22644  qustgplem  22656  prdstmdd  22659  tsmsadd  22682  cnmpt2vsca  22730  txmetcn  23085  cnmpt2ds  23378  fsum2cn  23406  cnmpopc  23459  htpyco2  23510  phtpyco2  23521  cnmpt2ip  23778  limccnp2  24417  dvcnp2  24444  dvaddbr  24462  dvmulbr  24463  dvcobr  24470  lhop1lem  24537  taylthlem2  24889  cxpcn3  25256  tpr2tp  31046  txsconnlem  32384  txsconn  32385  cvmlift2lem11  32457  cvmlift2lem12  32458
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