Theorem txcld 23949

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

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
2	1	cldss 23374	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝐴 ⊆ ∪ 𝑅)
3		eqid 2763	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
4	3	cldss 23374	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝐵 ⊆ ∪ 𝑆)
5		xpss12 5670	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑅 ∧ 𝐵 ⊆ ∪ 𝑆) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
6	2, 4, 5	syl2an 608	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
7		cldrcl 23371	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8		cldrcl 23371	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
9	1, 3	txuni 23938	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
10	7, 8, 9	syl2an 608	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
11	6, 10	sseqtrd 3949	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×t 𝑆))
12		difxp 6166	. . . 4 ⊢ ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)))
13	10	difeq1d 4065	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
14	12, 13	eqtr3id 2812	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15		txtop 23915	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
16	7, 8, 15	syl2an 608	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
17	7	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
18	8	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
19	1	cldopn 23376	. . . . . 6 ⊢ (𝐴 ∈ (Clsd‘𝑅) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
20	19	adantr 486	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
21	3	topopn 23251	. . . . . 6 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
22	18, 21	syl 18	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑆 ∈ 𝑆)
23		txopn 23948	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ((∪ 𝑅 ∖ 𝐴) ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
24	17, 18, 20, 22, 23	syl22anc 852	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
25	1	topopn 23251	. . . . . 6 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
26	17, 25	syl 18	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑅 ∈ 𝑅)
27	3	cldopn 23376	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
28	27	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
29		txopn 23948	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
30	17, 18, 26, 28, 29	syl22anc 852	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
31		unopn 23248	. . . 4 ⊢ (((𝑅 ×t 𝑆) ∈ Top ∧ ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆) ∧ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) ∈ (𝑅 ×t 𝑆))
32	16, 24, 30, 31	syl3anc 1399	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) ∈ (𝑅 ×t 𝑆))
33	14, 32	eqeltrrd 2865	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34		eqid 2763	. . . 4 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
35	34	iscld 23372	. . 3 ⊢ ((𝑅 ×t 𝑆) ∈ Top → ((𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)) ↔ ((𝐴 × 𝐵) ⊆ ∪ (𝑅 ×t 𝑆) ∧ (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))))
36	16, 35	syl 18	. 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))))
37	11, 33, 36	mpbir2and 726	1 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∪ cuni 4870   × cxp 5653  ‘cfv 6555  (class class class)co 7440  Topctop 23238  Clsdccld 23361   ×_t ctx 23906

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-topgen 17640  df-top 23239  df-topon 23256  df-bases 23291  df-cld 23364  df-tx 23908

This theorem is used by:  txcls  23950  cnmpopc  25276  sxbrsigalem3  34971