Theorem txcld 23771

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 2762	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
2	1	cldss 23197	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝐴 ⊆ ∪ 𝑅)
3		eqid 2762	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
4	3	cldss 23197	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝐵 ⊆ ∪ 𝑆)
5		xpss12 5675	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑅 ∧ 𝐵 ⊆ ∪ 𝑆) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
6	2, 4, 5	syl2an 607	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
7		cldrcl 23194	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8		cldrcl 23194	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
9	1, 3	txuni 23760	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
10	7, 8, 9	syl2an 607	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
11	6, 10	sseqtrd 3972	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×t 𝑆))
12		difxp 6160	. . . 4 ⊢ ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)))
13	10	difeq1d 4079	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
14	12, 13	eqtr3id 2811	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15		txtop 23737	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
16	7, 8, 15	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
17	7	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
18	8	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
19	1	cldopn 23199	. . . . . 6 ⊢ (𝐴 ∈ (Clsd‘𝑅) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
20	19	adantr 485	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
21	3	topopn 23074	. . . . . 6 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
22	18, 21	syl 18	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑆 ∈ 𝑆)
23		txopn 23770	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ((∪ 𝑅 ∖ 𝐴) ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
24	17, 18, 20, 22, 23	syl22anc 851	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
25	1	topopn 23074	. . . . . 6 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
26	17, 25	syl 18	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑅 ∈ 𝑅)
27	3	cldopn 23199	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
28	27	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
29		txopn 23770	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
30	17, 18, 26, 28, 29	syl22anc 851	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
31		unopn 23071	. . . 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 1397	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) ∈ (𝑅 ×t 𝑆))
33	14, 32	eqeltrrd 2863	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34		eqid 2762	. . . 4 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
35	34	iscld 23195	. . 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 725	1 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  ∪ cuni 4871   × cxp 5658  ‘cfv 6536  (class class class)co 7412  Topctop 23061  Clsdccld 23184   ×_t ctx 23728

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7984  df-2nd 7985  df-topgen 17502  df-top 23062  df-topon 23079  df-bases 23114  df-cld 23187  df-tx 23730

This theorem is used by:  txcls  23772  cnmpopc  25098  sxbrsigalem3  34671