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Theorem txcld 21627
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 2771 . . . . 5 𝑅 = 𝑅
21cldss 21054 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝐴 𝑅)
3 eqid 2771 . . . . 5 𝑆 = 𝑆
43cldss 21054 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝐵 𝑆)
5 xpss12 5264 . . . 4 ((𝐴 𝑅𝐵 𝑆) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
62, 4, 5syl2an 583 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
7 cldrcl 21051 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8 cldrcl 21051 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
91, 3txuni 21616 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
107, 8, 9syl2an 583 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
116, 10sseqtrd 3790 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆))
12 difxp 5699 . . . 4 (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵)))
1310difeq1d 3878 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
1412, 13syl5eqr 2819 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15 txtop 21593 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
167, 8, 15syl2an 583 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
177adantr 466 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
188adantl 467 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
191cldopn 21056 . . . . . 6 (𝐴 ∈ (Clsd‘𝑅) → ( 𝑅𝐴) ∈ 𝑅)
2019adantr 466 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅𝐴) ∈ 𝑅)
213topopn 20931 . . . . . 6 (𝑆 ∈ Top → 𝑆𝑆)
2218, 21syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆𝑆)
23 txopn 21626 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (( 𝑅𝐴) ∈ 𝑅 𝑆𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
2417, 18, 20, 22, 23syl22anc 1477 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
251topopn 20931 . . . . . 6 (𝑅 ∈ Top → 𝑅𝑅)
2617, 25syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅𝑅)
273cldopn 21056 . . . . . 6 (𝐵 ∈ (Clsd‘𝑆) → ( 𝑆𝐵) ∈ 𝑆)
2827adantl 467 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑆𝐵) ∈ 𝑆)
29 txopn 21626 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ( 𝑅𝑅 ∧ ( 𝑆𝐵) ∈ 𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
3017, 18, 26, 28, 29syl22anc 1477 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
31 unopn 20928 . . . 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 1476 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) ∈ (𝑅 ×t 𝑆))
3314, 32eqeltrrd 2851 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34 eqid 2771 . . . 4 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
3534iscld 21052 . . 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 692 1 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cdif 3720  cun 3721  wss 3723   cuni 4574   × cxp 5247  cfv 6031  (class class class)co 6793  Topctop 20918  Clsdccld 21041   ×t ctx 21584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-topgen 16312  df-top 20919  df-topon 20936  df-bases 20971  df-cld 21044  df-tx 21586
This theorem is referenced by:  txcls  21628  cnmpt2pc  22947  sxbrsigalem3  30674
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