Theorem txcmp 22794

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 22546	. . 3 ⊢ (𝑅 ∈ Comp → 𝑅 ∈ Top)
2		cmptop 22546	. . 3 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
3		txtop 22720	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5		eqid 2738	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2738	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
7		simpll 764	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑅 ∈ Comp)
8		simplr 766	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑆 ∈ Comp)
9		elpwi 4542	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
10	9	ad2antrl 725	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
11	5, 6	txuni 22743	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
12	1, 2, 11	syl2an 596	. . . . . . . 8 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
13	12	adantr 481	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
14		simprr 770	. . . . . . 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 22793	. . . . 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 3226	. . . . 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 457	. . 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 3103	. 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 22539	. 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 583	1 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∩ cin 3886   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839   × cxp 5587  (class class class)co 7275  Fincfn 8733  Topctop 22042  Compccmp 22537   ×_t ctx 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-fin 8737  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cmp 22538  df-tx 22713

This theorem is referenced by:  txcmpb  22795  txkgen  22803  ptcmpfi  22964  xkohmeo  22966  cnheiborlem  24117