Theorem txcld 22205

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 2821	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
2	1	cldss 21631	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝐴 ⊆ ∪ 𝑅)
3		eqid 2821	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
4	3	cldss 21631	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝐵 ⊆ ∪ 𝑆)
5		xpss12 5564	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑅 ∧ 𝐵 ⊆ ∪ 𝑆) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
6	2, 4, 5	syl2an 597	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
7		cldrcl 21628	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8		cldrcl 21628	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
9	1, 3	txuni 22194	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
10	7, 8, 9	syl2an 597	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
11	6, 10	sseqtrd 4006	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×t 𝑆))
12		difxp 6015	. . . 4 ⊢ ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)))
13	10	difeq1d 4097	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
14	12, 13	syl5eqr 2870	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15		txtop 22171	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
16	7, 8, 15	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
17	7	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
18	8	adantl 484	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
19	1	cldopn 21633	. . . . . 6 ⊢ (𝐴 ∈ (Clsd‘𝑅) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
20	19	adantr 483	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
21	3	topopn 21508	. . . . . 6 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
22	18, 21	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑆 ∈ 𝑆)
23		txopn 22204	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ((∪ 𝑅 ∖ 𝐴) ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
24	17, 18, 20, 22, 23	syl22anc 836	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
25	1	topopn 21508	. . . . . 6 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
26	17, 25	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑅 ∈ 𝑅)
27	3	cldopn 21633	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
28	27	adantl 484	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
29		txopn 22204	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
30	17, 18, 26, 28, 29	syl22anc 836	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
31		unopn 21505	. . . 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 1367	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) ∈ (𝑅 ×t 𝑆))
33	14, 32	eqeltrrd 2914	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34		eqid 2821	. . . 4 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
35	34	iscld 21629	. . 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 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))))
37	11, 33, 36	mpbir2and 711	1 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ∖ cdif 3932   ∪ cun 3933   ⊆ wss 3935  ∪ cuni 4831   × cxp 5547  ‘cfv 6349  (class class class)co 7150  Topctop 21495  Clsdccld 21618   ×_t ctx 22162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-topgen 16711  df-top 21496  df-topon 21513  df-bases 21548  df-cld 21621  df-tx 22164

This theorem is referenced by:  txcls  22206  cnmpopc  23526  sxbrsigalem3  31525