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Theorem txopn 23626
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 2735 . . . . . 6 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txbasex 23590 . . . . 5 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3 bastg 22989 . . . . 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 2735 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5703 . . . . . . . 8 (𝑢 = 𝐴 → (𝑢 × 𝑣) = (𝐴 × 𝑣))
87eqeq2d 2746 . . . . . . 7 (𝑢 = 𝐴 → ((𝐴 × 𝐵) = (𝑢 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑣)))
9 xpeq2 5710 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴 × 𝑣) = (𝐴 × 𝐵))
109eqeq2d 2746 . . . . . . 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 1449 . . . . 5 ((𝐴𝑅𝐵𝑆) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
13 xpexg 7769 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ V)
14 eqid 2735 . . . . . . 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 3996 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
201txval 23588 . . 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 2842 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  wss 3963   × cxp 5687  ran crn 5690  cfv 6563  (class class class)co 7431  cmpo 7433  topGenctg 17484   ×t ctx 23584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-topgen 17490  df-tx 23586
This theorem is referenced by:  txcld  23627  txbasval  23630  neitx  23631  tx1cn  23633  tx2cn  23634  txlly  23660  txnlly  23661  txhaus  23671  txlm  23672  tx1stc  23674  txkgen  23676  xkococnlem  23683  cxpcn3  26806  cvmlift2lem11  35298  cvmlift2lem12  35299
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