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Theorem txtop 22720
Description: The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
txtop ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)

Proof of Theorem txtop
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2738 . . 3 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txval 22715 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
3 topbas 22122 . . . 4 (𝑅 ∈ Top → 𝑅 ∈ TopBases)
4 topbas 22122 . . . 4 (𝑆 ∈ Top → 𝑆 ∈ TopBases)
51txbas 22718 . . . 4 ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases)
63, 4, 5syl2an 596 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases)
7 tgcl 22119 . . 3 (ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases → (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))) ∈ Top)
86, 7syl 17 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))) ∈ Top)
92, 8eqeltrd 2839 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106   × cxp 5587  ran crn 5590  cfv 6433  (class class class)co 7275  cmpo 7277  topGenctg 17148  Topctop 22042  TopBasesctb 22095   ×t ctx 22711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-topgen 17154  df-top 22043  df-bases 22096  df-tx 22713
This theorem is referenced by:  txtopi  22741  txtopon  22742  txcld  22754  neitx  22758  txlly  22787  txnlly  22788  txcmplem1  22792  txcmp  22794  hausdiag  22796  txhaus  22798  tx1stc  22801  txkgen  22803  xkococn  22811  xkoinjcn  22838  txconn  22840  imasnopn  22841  imasncls  22843  utop2nei  23402  utop3cls  23403  qtophaus  31786  txpconn  33194
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