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Theorem txcmp 22540
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.
StepHypRef Expression
1 cmptop 22292 . . 3 (𝑅 ∈ Comp → 𝑅 ∈ Top)
2 cmptop 22292 . . 3 (𝑆 ∈ Comp → 𝑆 ∈ Top)
3 txtop 22466 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 599 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2737 . . . . . 6 𝑅 = 𝑅
6 eqid 2737 . . . . . 6 𝑆 = 𝑆
7 simpll 767 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑅 ∈ Comp)
8 simplr 769 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑆 ∈ Comp)
9 elpwi 4522 . . . . . . 7 (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
109ad2antrl 728 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
115, 6txuni 22489 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
121, 2, 11syl2an 599 . . . . . . . 8 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1312adantr 484 . . . . . . 7 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
14 simprr 773 . . . . . . 7 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (𝑅 ×t 𝑆) = 𝑤)
1513, 14eqtrd 2777 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ( 𝑅 × 𝑆) = 𝑤)
165, 6, 7, 8, 10, 15txcmplem2 22539 . . . . 5 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)( 𝑅 × 𝑆) = 𝑣)
1713eqeq1d 2739 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (( 𝑅 × 𝑆) = 𝑣 (𝑅 ×t 𝑆) = 𝑣))
1817rexbidv 3216 . . . . 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))
1916, 18mpbid 235 . . . 4 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣)
2019expr 460 . . 3 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ 𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)) → ( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
2120ralrimiva 3105 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
22 eqid 2737 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2322iscmp 22285 . 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)))
244, 21, 23sylanbrc 586 1 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cin 3865  wss 3866  𝒫 cpw 4513   cuni 4819   × cxp 5549  (class class class)co 7213  Fincfn 8626  Topctop 21790  Compccmp 22283   ×t ctx 22457
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-1o 8202  df-er 8391  df-en 8627  df-dom 8628  df-fin 8630  df-topgen 16948  df-top 21791  df-topon 21808  df-bases 21843  df-cmp 22284  df-tx 22459
This theorem is referenced by:  txcmpb  22541  txkgen  22549  ptcmpfi  22710  xkohmeo  22712  cnheiborlem  23851
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