Theorem txcld 23627

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 2735	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
2	1	cldss 23053	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝐴 ⊆ ∪ 𝑅)
3		eqid 2735	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
4	3	cldss 23053	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝐵 ⊆ ∪ 𝑆)
5		xpss12 5704	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑅 ∧ 𝐵 ⊆ ∪ 𝑆) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
6	2, 4, 5	syl2an 596	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
7		cldrcl 23050	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8		cldrcl 23050	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
9	1, 3	txuni 23616	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
10	7, 8, 9	syl2an 596	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
11	6, 10	sseqtrd 4036	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×t 𝑆))
12		difxp 6186	. . . 4 ⊢ ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)))
13	10	difeq1d 4135	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
14	12, 13	eqtr3id 2789	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15		txtop 23593	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
16	7, 8, 15	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
17	7	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
18	8	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
19	1	cldopn 23055	. . . . . 6 ⊢ (𝐴 ∈ (Clsd‘𝑅) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
20	19	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
21	3	topopn 22928	. . . . . 6 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
22	18, 21	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑆 ∈ 𝑆)
23		txopn 23626	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ((∪ 𝑅 ∖ 𝐴) ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
24	17, 18, 20, 22, 23	syl22anc 839	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
25	1	topopn 22928	. . . . . 6 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
26	17, 25	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑅 ∈ 𝑅)
27	3	cldopn 23055	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
28	27	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
29		txopn 23626	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
30	17, 18, 26, 28, 29	syl22anc 839	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
31		unopn 22925	. . . 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 1370	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) ∈ (𝑅 ×t 𝑆))
33	14, 32	eqeltrrd 2840	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34		eqid 2735	. . . 4 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
35	34	iscld 23051	. . 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 713	1 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106   ∖ cdif 3960   ∪ cun 3961   ⊆ wss 3963  ∪ cuni 4912   × cxp 5687  ‘cfv 6563  (class class class)co 7431  Topctop 22915  Clsdccld 23040   ×_t ctx 23584

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-topgen 17490  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-tx 23586

This theorem is referenced by:  txcls  23628  cnmpopc  24969  sxbrsigalem3  34254