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Theorem txcmp 23129
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 22881 . . 3 (𝑅 ∈ Comp → 𝑅 ∈ Top)
2 cmptop 22881 . . 3 (𝑆 ∈ Comp → 𝑆 ∈ Top)
3 txtop 23055 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 597 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2733 . . . . . 6 𝑅 = 𝑅
6 eqid 2733 . . . . . 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 4608 . . . . . . 7 (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
109ad2antrl 727 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
115, 6txuni 23078 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
121, 2, 11syl2an 597 . . . . . . . 8 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1312adantr 482 . . . . . . 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)
1513, 14eqtrd 2773 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ( 𝑅 × 𝑆) = 𝑤)
165, 6, 7, 8, 10, 15txcmplem2 23128 . . . . 5 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)( 𝑅 × 𝑆) = 𝑣)
1713eqeq1d 2735 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (( 𝑅 × 𝑆) = 𝑣 (𝑅 ×t 𝑆) = 𝑣))
1817rexbidv 3179 . . . . 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 231 . . . 4 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣)
2019expr 458 . . 3 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ 𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)) → ( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
2120ralrimiva 3147 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
22 eqid 2733 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2322iscmp 22874 . 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 584 1 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  cin 3946  wss 3947  𝒫 cpw 4601   cuni 4907   × cxp 5673  (class class class)co 7404  Fincfn 8935  Topctop 22377  Compccmp 22872   ×t ctx 23046
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-1o 8461  df-er 8699  df-en 8936  df-dom 8937  df-fin 8939  df-topgen 17385  df-top 22378  df-topon 22395  df-bases 22431  df-cmp 22873  df-tx 23048
This theorem is referenced by:  txcmpb  23130  txkgen  23138  ptcmpfi  23299  xkohmeo  23301  cnheiborlem  24452
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