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Theorem txopn 23550
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 2725 . . . . . 6 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txbasex 23514 . . . . 5 ((𝑅𝑉𝑆𝑊) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3 bastg 22913 . . . . 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 479 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
6 eqid 2725 . . . . . 6 (𝐴 × 𝐵) = (𝐴 × 𝐵)
7 xpeq1 5692 . . . . . . . 8 (𝑢 = 𝐴 → (𝑢 × 𝑣) = (𝐴 × 𝑣))
87eqeq2d 2736 . . . . . . 7 (𝑢 = 𝐴 → ((𝐴 × 𝐵) = (𝑢 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑣)))
9 xpeq2 5699 . . . . . . . 8 (𝑣 = 𝐵 → (𝐴 × 𝑣) = (𝐴 × 𝐵))
109eqeq2d 2736 . . . . . . 7 (𝑣 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
118, 10rspc2ev 3619 . . . . . 6 ((𝐴𝑅𝐵𝑆 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
126, 11mp3an3 1446 . . . . 5 ((𝐴𝑅𝐵𝑆) → ∃𝑢𝑅𝑣𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
13 xpexg 7753 . . . . . 6 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ V)
14 eqid 2725 . . . . . . 7 (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
1514elrnmpog 7556 . . . . . 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 256 . . . 4 ((𝐴𝑅𝐵𝑆) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
1817adantl 480 . . 3 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
195, 18sseldd 3977 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
201txval 23512 . . 3 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2120adantr 479 . 2 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
2219, 21eleqtrrd 2828 1 (((𝑅𝑉𝑆𝑊) ∧ (𝐴𝑅𝐵𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059  Vcvv 3461  wss 3944   × cxp 5676  ran crn 5679  cfv 6549  (class class class)co 7419  cmpo 7421  topGenctg 17422   ×t ctx 23508
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-topgen 17428  df-tx 23510
This theorem is referenced by:  txcld  23551  txbasval  23554  neitx  23555  tx1cn  23557  tx2cn  23558  txlly  23584  txnlly  23585  txhaus  23595  txlm  23596  tx1stc  23598  txkgen  23600  xkococnlem  23607  cxpcn3  26728  cvmlift2lem11  35054  cvmlift2lem12  35055
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