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Theorem txopn 23610
Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
txopn (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))

Proof of Theorem txopn
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . . 6 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txbasex 23574 . . . . 5 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3 bastg 22973 . . . . 5 (ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
42, 3syl 17 . . . 4 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
54adantr 480 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
6 eqid 2737 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5699 . . . . . . . 8 (𝑢 = 𝐴 → (𝑢 × 𝑣) = (𝐴 × 𝑣))
87eqeq2d 2748 . . . . . . 7 (𝑢 = 𝐴 → ((𝐴 × 𝐵) = (𝑢 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑣)))
9 xpeq2 5706 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴 × 𝑣) = (𝐴 × 𝐵))
109eqeq2d 2748 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3635 . . . . . 6 ((𝐴𝑅𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
126, 11mp3an3 1452 . . . . 5 ((𝐴𝑅𝐵𝑆) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
13 xpexg 7770 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ V)
14 eqid 2737 . . . . . . 7 (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
1514elrnmpog 7568 . . . . . 6 ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ↔ ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣)))
1613, 15syl 17 . . . . 5 ((𝐴𝑅𝐵𝑆) → ((𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ↔ ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣)))
1712, 16mpbird 257 . . . 4 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
1817adantl 481 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
195, 18sseldd 3984 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
201txval 23572 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2120adantr 480 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2219, 21eleqtrrd 2844 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  wss 3951   × cxp 5683  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  topGenctg 17482   ×t ctx 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-topgen 17488  df-tx 23570
This theorem is referenced by:  txcld  23611  txbasval  23614  neitx  23615  tx1cn  23617  tx2cn  23618  txlly  23644  txnlly  23645  txhaus  23655  txlm  23656  tx1stc  23658  txkgen  23660  xkococnlem  23667  cxpcn3  26791  cvmlift2lem11  35318  cvmlift2lem12  35319
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