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Theorem txcld 23611
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 2737 . . . . 5 𝑅 = 𝑅
21cldss 23037 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝐴 𝑅)
3 eqid 2737 . . . . 5 𝑆 = 𝑆
43cldss 23037 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝐵 𝑆)
5 xpss12 5700 . . . 4 ((𝐴 𝑅𝐵 𝑆) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
62, 4, 5syl2an 596 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
7 cldrcl 23034 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8 cldrcl 23034 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
91, 3txuni 23600 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
107, 8, 9syl2an 596 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
116, 10sseqtrd 4020 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆))
12 difxp 6184 . . . 4 (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵)))
1310difeq1d 4125 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
1412, 13eqtr3id 2791 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15 txtop 23577 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
167, 8, 15syl2an 596 . . . 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 23039 . . . . . 6 (𝐴 ∈ (Clsd‘𝑅) → ( 𝑅𝐴) ∈ 𝑅)
2019adantr 480 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅𝐴) ∈ 𝑅)
213topopn 22912 . . . . . 6 (𝑆 ∈ Top → 𝑆𝑆)
2218, 21syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆𝑆)
23 txopn 23610 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (( 𝑅𝐴) ∈ 𝑅 𝑆𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
2417, 18, 20, 22, 23syl22anc 839 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
251topopn 22912 . . . . . 6 (𝑅 ∈ Top → 𝑅𝑅)
2617, 25syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅𝑅)
273cldopn 23039 . . . . . 6 (𝐵 ∈ (Clsd‘𝑆) → ( 𝑆𝐵) ∈ 𝑆)
2827adantl 481 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑆𝐵) ∈ 𝑆)
29 txopn 23610 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ( 𝑅𝑅 ∧ ( 𝑆𝐵) ∈ 𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
3017, 18, 26, 28, 29syl22anc 839 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
31 unopn 22909 . . . 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 1373 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) ∈ (𝑅 ×t 𝑆))
3314, 32eqeltrrd 2842 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34 eqid 2737 . . . 4 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
3534iscld 23035 . . 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 713 1 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cdif 3948  cun 3949  wss 3951   cuni 4907   × cxp 5683  cfv 6561  (class class class)co 7431  Topctop 22899  Clsdccld 23024   ×t ctx 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-tx 23570
This theorem is referenced by:  txcls  23612  cnmpopc  24955  sxbrsigalem3  34274
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