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Theorem txcmp 21660
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 21412 . . 3 (𝑅 ∈ Comp → 𝑅 ∈ Top)
2 cmptop 21412 . . 3 (𝑆 ∈ Comp → 𝑆 ∈ Top)
3 txtop 21586 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 585 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2806 . . . . . 6 𝑅 = 𝑅
6 eqid 2806 . . . . . 6 𝑆 = 𝑆
7 simpll 774 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑅 ∈ Comp)
8 simplr 776 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑆 ∈ Comp)
9 elpwi 4361 . . . . . . 7 (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
109ad2antrl 710 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
115, 6txuni 21609 . . . . . . . . 9 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
121, 2, 11syl2an 585 . . . . . . . 8 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
1312adantr 468 . . . . . . 7 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ( 𝑅 × 𝑆) = (𝑅 ×t 𝑆))
14 simprr 780 . . . . . . 7 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (𝑅 ×t 𝑆) = 𝑤)
1513, 14eqtrd 2840 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ( 𝑅 × 𝑆) = 𝑤)
165, 6, 7, 8, 10, 15txcmplem2 21659 . . . . 5 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)( 𝑅 × 𝑆) = 𝑣)
1713eqeq1d 2808 . . . . . 6 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → (( 𝑅 × 𝑆) = 𝑣 (𝑅 ×t 𝑆) = 𝑣))
1817rexbidv 3240 . . . . 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 223 . . . 4 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ (𝑅 ×t 𝑆) = 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣)
2019expr 446 . . 3 (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ 𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)) → ( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
2120ralrimiva 3154 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)( (𝑅 ×t 𝑆) = 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin) (𝑅 ×t 𝑆) = 𝑣))
22 eqid 2806 . . 3 (𝑅 ×t 𝑆) = (𝑅 ×t 𝑆)
2322iscmp 21405 . 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 574 1 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  wral 3096  wrex 3097  cin 3768  wss 3769  𝒫 cpw 4351   cuni 4630   × cxp 5309  (class class class)co 6874  Fincfn 8192  Topctop 20911  Compccmp 21403   ×t ctx 21577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-iin 4715  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-en 8193  df-dom 8194  df-fin 8196  df-topgen 16309  df-top 20912  df-topon 20929  df-bases 20964  df-cmp 21404  df-tx 21579
This theorem is referenced by:  txcmpb  21661  txkgen  21669  ptcmpfi  21830  xkohmeo  21832  cnheiborlem  22966
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