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Theorem txtop 22176
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 2821 . . 3 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
21txval 22171 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
3 topbas 21579 . . . 4 (𝑅 ∈ Top → 𝑅 ∈ TopBases)
4 topbas 21579 . . . 4 (𝑆 ∈ Top → 𝑆 ∈ TopBases)
51txbas 22174 . . . 4 ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases)
63, 4, 5syl2an 597 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases)
7 tgcl 21576 . . 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 2913 1 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2110   × cxp 5552  ran crn 5555  cfv 6354  (class class class)co 7155  cmpo 7157  topGenctg 16710  Topctop 21500  TopBasesctb 21552   ×t ctx 22167
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-topgen 16716  df-top 21501  df-bases 21553  df-tx 22169
This theorem is referenced by:  txtopi  22197  txtopon  22198  txcld  22210  neitx  22214  txlly  22243  txnlly  22244  txcmplem1  22248  txcmp  22250  hausdiag  22252  txhaus  22254  tx1stc  22257  txkgen  22259  xkococn  22267  xkoinjcn  22294  txconn  22296  imasnopn  22297  imasncls  22299  utop2nei  22858  utop3cls  22859  qtophaus  31100  txpconn  32479
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