Theorem txcmp 23672

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 23424	. . 3 ⊢ (𝑅 ∈ Comp → 𝑅 ∈ Top)
2		cmptop 23424	. . 3 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
3		txtop 23598	. . 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 2740	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2740	. . . . . 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 4629	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
10	9	ad2antrl 727	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
11	5, 6	txuni 23621	. . . . . . . . 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 772	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)
15	13, 14	eqtrd 2780	. . . . . 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 23671	. . . . 5 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)(∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣)
17	13	eqeq1d 2742	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ((∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣 ↔ ∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
18	17	rexbidv 3185	. . . . 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 3152	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)(∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
22		eqid 2740	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
23	22	iscmp 23417	. 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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076   ∩ cin 3975   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   × cxp 5698  (class class class)co 7448  Fincfn 9003  Topctop 22920  Compccmp 23415   ×_t ctx 23589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-iin 5018  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-1o 8522  df-2o 8523  df-en 9004  df-dom 9005  df-fin 9007  df-topgen 17503  df-top 22921  df-topon 22938  df-bases 22974  df-cmp 23416  df-tx 23591

This theorem is referenced by:  txcmpb  23673  txkgen  23681  ptcmpfi  23842  xkohmeo  23844  cnheiborlem  25005