MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txcld Structured version   Visualization version   GIF version

Theorem txcld 21216
Description: The product of two closed sets is closed in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
txcld ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

Proof of Theorem txcld
StepHypRef Expression
1 eqid 2610 . . . . 5 𝑅 = 𝑅
21cldss 20643 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝐴 𝑅)
3 eqid 2610 . . . . 5 𝑆 = 𝑆
43cldss 20643 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝐵 𝑆)
5 xpss12 5148 . . . 4 ((𝐴 𝑅𝐵 𝑆) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
62, 4, 5syl2an 493 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
7 cldrcl 20640 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8 cldrcl 20640 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
91, 3txuni 21205 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
107, 8, 9syl2an 493 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
116, 10sseqtrd 3604 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆))
12 difxp 5477 . . . 4 (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵)))
1310difeq1d 3689 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
1412, 13syl5eqr 2658 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15 txtop 21182 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
167, 8, 15syl2an 493 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
177adantr 480 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
188adantl 481 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
191cldopn 20645 . . . . . 6 (𝐴 ∈ (Clsd‘𝑅) → ( 𝑅𝐴) ∈ 𝑅)
2019adantr 480 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅𝐴) ∈ 𝑅)
213topopn 20536 . . . . . 6 (𝑆 ∈ Top → 𝑆𝑆)
2218, 21syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆𝑆)
23 txopn 21215 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (( 𝑅𝐴) ∈ 𝑅 𝑆𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
2417, 18, 20, 22, 23syl22anc 1319 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
251topopn 20536 . . . . . 6 (𝑅 ∈ Top → 𝑅𝑅)
2617, 25syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅𝑅)
273cldopn 20645 . . . . . 6 (𝐵 ∈ (Clsd‘𝑆) → ( 𝑆𝐵) ∈ 𝑆)
2827adantl 481 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑆𝐵) ∈ 𝑆)
29 txopn 21215 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ( 𝑅𝑅 ∧ ( 𝑆𝐵) ∈ 𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
3017, 18, 26, 28, 29syl22anc 1319 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
31 unopn 20533 . . . 4 (((𝑅 ×t 𝑆) ∈ Top ∧ (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆) ∧ ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) ∈ (𝑅 ×t 𝑆))
3216, 24, 30, 31syl3anc 1318 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) ∈ (𝑅 ×t 𝑆))
3314, 32eqeltrrd 2689 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34 eqid 2610 . . . 4 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
3534iscld 20641 . . 3 ((𝑅 ×t 𝑆) ∈ Top → ((𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)) ↔ ((𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆) ∧ ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))))
3616, 35syl 17 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)) ↔ ((𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆) ∧ ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))))
3711, 33, 36mpbir2and 959 1 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cdif 3537  cun 3538  wss 3540   cuni 4372   × cxp 5036  cfv 5804  (class class class)co 6549  Topctop 20517  Clsdccld 20630   ×t ctx 21173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-tx 21175
This theorem is referenced by:  txcls  21217  cnmpt2pc  22535  sxbrsigalem3  29661
  Copyright terms: Public domain W3C validator