Theorem txcmp 22702

Description: The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)

Assertion

Ref	Expression
txcmp	⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)

Proof of Theorem txcmp

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmptop 22454	. . 3 ⊢ (𝑅 ∈ Comp → 𝑅 ∈ Top)
2		cmptop 22454	. . 3 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
3		txtop 22628	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5		eqid 2738	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2738	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
7		simpll 763	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑅 ∈ Comp)
8		simplr 765	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑆 ∈ Comp)
9		elpwi 4539	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
10	9	ad2antrl 724	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
11	5, 6	txuni 22651	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
12	1, 2, 11	syl2an 595	. . . . . . . 8 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
13	12	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
14		simprr 769	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)
15	13, 14	eqtrd 2778	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∪ 𝑅 × ∪ 𝑆) = ∪ 𝑤)
16	5, 6, 7, 8, 10, 15	txcmplem2 22701	. . . . 5 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)(∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣)
17	13	eqeq1d 2740	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ((∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣 ↔ ∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
18	17	rexbidv 3225	. . . . 5 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)(∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
19	16, 18	mpbid 231	. . . 4 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣)
20	19	expr 456	. . 3 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ 𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)) → (∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
21	20	ralrimiva 3107	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)(∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
22		eqid 2738	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
23	22	iscmp 22447	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ Comp ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)(∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣)))
24	4, 21, 23	sylanbrc 582	1 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4530  ∪ cuni 4836   × cxp 5578  (class class class)co 7255  Fincfn 8691  Topctop 21950  Compccmp 22445   ×_t ctx 22619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-fin 8695  df-topgen 17071  df-top 21951  df-topon 21968  df-bases 22004  df-cmp 22446  df-tx 22621

This theorem is referenced by:  txcmpb  22703  txkgen  22711  ptcmpfi  22872  xkohmeo  22874  cnheiborlem  24023