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Theorem txcmp 23651
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 23403 . . 3 (𝑅 ∈ Comp → 𝑅 ∈ Top)
2 cmptop 23403 . . 3 (𝑆 ∈ Comp → 𝑆 ∈ Top)
3 txtop 23577 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 596 . 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 4607 . . . . . . 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 23600 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
121, 2, 11syl2an 596 . . . . . . . 8 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1312adantr 480 . . . . . . 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 23650 . . . . 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 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 232 . . . 4 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣)
2019expr 456 . . 3 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ 𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)) → ( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
2120ralrimiva 3146 . 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 23396 . 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 583 1 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600   cuni 4907   × cxp 5683  (class class class)co 7431  Fincfn 8985  Topctop 22899  Compccmp 23394   ×t ctx 23568
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-en 8986  df-dom 8987  df-fin 8989  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cmp 23395  df-tx 23570
This theorem is referenced by:  txcmpb  23652  txkgen  23660  ptcmpfi  23821  xkohmeo  23823  cnheiborlem  24986
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