Theorem txcld 21777

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 2825	. . . . 5 ⊢ ∪ 𝑅 = ∪ 𝑅
2	1	cldss 21204	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝐴 ⊆ ∪ 𝑅)
3		eqid 2825	. . . . 5 ⊢ ∪ 𝑆 = ∪ 𝑆
4	3	cldss 21204	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝐵 ⊆ ∪ 𝑆)
5		xpss12 5357	. . . 4 ⊢ ((𝐴 ⊆ ∪ 𝑅 ∧ 𝐵 ⊆ ∪ 𝑆) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
6	2, 4, 5	syl2an 589	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ (∪ 𝑅 × ∪ 𝑆))
7		cldrcl 21201	. . . 4 ⊢ (𝐴 ∈ (Clsd‘𝑅) → 𝑅 ∈ Top)
8		cldrcl 21201	. . . 4 ⊢ (𝐵 ∈ (Clsd‘𝑆) → 𝑆 ∈ Top)
9	1, 3	txuni 21766	. . . 4 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
10	7, 8, 9	syl2an 589	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
11	6, 10	sseqtrd 3866	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ⊆ ∪ (𝑅 ×t 𝑆))
12		difxp 5799	. . . 4 ⊢ ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)))
13	10	difeq1d 3954	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 × ∪ 𝑆) ∖ (𝐴 × 𝐵)) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
14	12, 13	syl5eqr 2875	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) = (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)))
15		txtop 21743	. . . . 5 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
16	7, 8, 15	syl2an 589	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝑅 ×t 𝑆) ∈ Top)
17	7	adantr 474	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑅 ∈ Top)
18	8	adantl 475	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → 𝑆 ∈ Top)
19	1	cldopn 21206	. . . . . 6 ⊢ (𝐴 ∈ (Clsd‘𝑅) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
20	19	adantr 474	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 ∖ 𝐴) ∈ 𝑅)
21	3	topopn 21081	. . . . . 6 ⊢ (𝑆 ∈ Top → ∪ 𝑆 ∈ 𝑆)
22	18, 21	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑆 ∈ 𝑆)
23		txopn 21776	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ ((∪ 𝑅 ∖ 𝐴) ∈ 𝑅 ∧ ∪ 𝑆 ∈ 𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
24	17, 18, 20, 22, 23	syl22anc 872	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∈ (𝑅 ×t 𝑆))
25	1	topopn 21081	. . . . . 6 ⊢ (𝑅 ∈ Top → ∪ 𝑅 ∈ 𝑅)
26	17, 25	syl 17	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → ∪ 𝑅 ∈ 𝑅)
27	3	cldopn 21206	. . . . . 6 ⊢ (𝐵 ∈ (Clsd‘𝑆) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
28	27	adantl 475	. . . . 5 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)
29		txopn 21776	. . . . 5 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (∪ 𝑅 ∈ 𝑅 ∧ (∪ 𝑆 ∖ 𝐵) ∈ 𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
30	17, 18, 26, 28, 29	syl22anc 872	. . . 4 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵)) ∈ (𝑅 ×t 𝑆))
31		unopn 21078	. . . 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 1494	. . 3 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (((∪ 𝑅 ∖ 𝐴) × ∪ 𝑆) ∪ (∪ 𝑅 × (∪ 𝑆 ∖ 𝐵))) ∈ (𝑅 ×t 𝑆))
33	14, 32	eqeltrrd 2907	. 2 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (∪ (𝑅 ×t 𝑆) ∖ (𝐴 × 𝐵)) ∈ (𝑅 ×t 𝑆))
34		eqid 2825	. . . 4 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
35	34	iscld 21202	. . 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 704	1 ⊢ ((𝐴 ∈ (Clsd‘𝑅) ∧ 𝐵 ∈ (Clsd‘𝑆)) → (𝐴 × 𝐵) ∈ (Clsd‘(𝑅 ×t 𝑆)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1656   ∈ wcel 2164   ∖ cdif 3795   ∪ cun 3796   ⊆ wss 3798  ∪ cuni 4658   × cxp 5340  ‘cfv 6123  (class class class)co 6905  Topctop 21068  Clsdccld 21191   ×_t ctx 21734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-1st 7428  df-2nd 7429  df-topgen 16457  df-top 21069  df-topon 21086  df-bases 21121  df-cld 21194  df-tx 21736

This theorem is referenced by:  txcls  21778  cnmpt2pc  23097  sxbrsigalem3  30868