Theorem txcmp 23564

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 23316	. . 3 ⊢ (𝑅 ∈ Comp → 𝑅 ∈ Top)
2		cmptop 23316	. . 3 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
3		txtop 23490	. . 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 2729	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2729	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
7		simpll 766	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑅 ∈ Comp)
8		simplr 768	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑆 ∈ Comp)
9		elpwi 4566	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
10	9	ad2antrl 728	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
11	5, 6	txuni 23513	. . . . . . . . 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 480	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
14		simprr 772	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)
15	13, 14	eqtrd 2764	. . . . . 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 23563	. . . . 5 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)(∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣)
17	13	eqeq1d 2731	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ((∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣 ↔ ∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
18	17	rexbidv 3157	. . . . 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 232	. . . 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 3125	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)(∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
22		eqid 2729	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
23	22	iscmp 23309	. 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 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3910   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867   × cxp 5629  (class class class)co 7369  Fincfn 8895  Topctop 22814  Compccmp 23307   ×_t ctx 23481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-1o 8411  df-2o 8412  df-en 8896  df-dom 8897  df-fin 8899  df-topgen 17383  df-top 22815  df-topon 22832  df-bases 22867  df-cmp 23308  df-tx 23483

This theorem is referenced by:  txcmpb  23565  txkgen  23573  ptcmpfi  23734  xkohmeo  23736  cnheiborlem  24887