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Theorem txcld 21608
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 2760 . . . . 5 𝑅 = 𝑅
21cldss 21035 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝐴 𝑅)
3 eqid 2760 . . . . 5 𝑆 = 𝑆
43cldss 21035 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝐵 𝑆)
5 xpss12 5281 . . . 4 ((𝐴 𝑅𝐵 𝑆) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
62, 4, 5syl2an 495 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ( 𝑅 × 𝑆))
7 cldrcl 21032 . . . 4 (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8 cldrcl 21032 . . . 4 (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
91, 3txuni 21597 . . . 4 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
107, 8, 9syl2an 495 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
116, 10sseqtrd 3782 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (𝑅 ×t 𝑆))
12 difxp 5716 . . . 4 (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵)))
1310difeq1d 3870 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅 × 𝑆) ∖ (𝐴 × 𝐵)) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
1412, 13syl5eqr 2808 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) = ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15 txtop 21574 . . . . 5 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
167, 8, 15syl2an 495 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
177adantr 472 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
188adantl 473 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
191cldopn 21037 . . . . . 6 (𝐴 ∈ (Clsd‘𝑅) → ( 𝑅𝐴) ∈ 𝑅)
2019adantr 472 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅𝐴) ∈ 𝑅)
213topopn 20913 . . . . . 6 (𝑆 ∈ Top → 𝑆𝑆)
2218, 21syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆𝑆)
23 txopn 21607 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (( 𝑅𝐴) ∈ 𝑅 𝑆𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
2417, 18, 20, 22, 23syl22anc 1478 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (( 𝑅𝐴) × 𝑆) ∈ (𝑅 ×t 𝑆))
251topopn 20913 . . . . . 6 (𝑅 ∈ Top → 𝑅𝑅)
2617, 25syl 17 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅𝑅)
273cldopn 21037 . . . . . 6 (𝐵 ∈ (Clsd‘𝑆) → ( 𝑆𝐵) ∈ 𝑆)
2827adantl 473 . . . . 5 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑆𝐵) ∈ 𝑆)
29 txopn 21607 . . . . 5 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ( 𝑅𝑅 ∧ ( 𝑆𝐵) ∈ 𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
3017, 18, 26, 28, 29syl22anc 1478 . . . 4 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( 𝑅 × ( 𝑆𝐵)) ∈ (𝑅 ×t 𝑆))
31 unopn 20910 . . . 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 1477 . . 3 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((( 𝑅𝐴) × 𝑆) ∪ ( 𝑅 × ( 𝑆𝐵))) ∈ (𝑅 ×t 𝑆))
3314, 32eqeltrrd 2840 . 2 ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ( (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34 eqid 2760 . . . 4 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
3534iscld 21033 . . 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 995 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 383   = wceq 1632  wcel 2139  cdif 3712  cun 3713  wss 3715   cuni 4588   × cxp 5264  cfv 6049  (class class class)co 6813  Topctop 20900  Clsdccld 21022   ×t ctx 21565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-tx 21567
This theorem is referenced by:  txcls  21609  cnmpt2pc  22928  sxbrsigalem3  30643
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