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Theorem txcmp 22289
 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 22041 . . 3 (𝑅 ∈ Comp → 𝑅 ∈ Top)
2 cmptop 22041 . . 3 (𝑆 ∈ Comp → 𝑆 ∈ Top)
3 txtop 22215 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 598 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2798 . . . . . 6 𝑅 = 𝑅
6 eqid 2798 . . . . . 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 4509 . . . . . . 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 22238 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
121, 2, 11syl2an 598 . . . . . . . 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 772 . . . . . . 7 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (𝑅 ×t 𝑆) = 𝑤)
1513, 14eqtrd 2833 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ( 𝑅 × 𝑆) = 𝑤)
165, 6, 7, 8, 10, 15txcmplem2 22288 . . . . 5 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)( 𝑅 × 𝑆) = 𝑣)
1713eqeq1d 2800 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (( 𝑅 × 𝑆) = 𝑣 (𝑅 ×t 𝑆) = 𝑣))
1817rexbidv 3257 . . . . 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 3149 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
22 eqid 2798 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2322iscmp 22034 . 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 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∩ cin 3882   ⊆ wss 3883  𝒫 cpw 4500  ∪ cuni 4804   × cxp 5521  (class class class)co 7145  Fincfn 8510  Topctop 21539  Compccmp 22032   ×t ctx 22206 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-iin 4888  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-oadd 8107  df-er 8290  df-en 8511  df-dom 8512  df-fin 8514  df-topgen 16729  df-top 21540  df-topon 21557  df-bases 21592  df-cmp 22033  df-tx 22208 This theorem is referenced by:  txcmpb  22290  txkgen  22298  ptcmpfi  22459  xkohmeo  22461  cnheiborlem  23600
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