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Theorem txcmpb 21357
Description: The topological product of two nonempty topologies is compact iff the component topologies are both compact. (Contributed by Mario Carneiro, 14-Sep-2014.)
Hypotheses
Ref Expression
txcmpb.1 𝑋 = 𝑅
txcmpb.2 𝑌 = 𝑆
Assertion
Ref Expression
txcmpb (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))

Proof of Theorem txcmpb
StepHypRef Expression
1 simpr 477 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
2 simplrr 800 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑌 ≠ ∅)
3 fo1stres 7137 . . . . . . 7 (𝑌 ≠ ∅ → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑋)
42, 3syl 17 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑋)
5 txcmpb.1 . . . . . . . . 9 𝑋 = 𝑅
6 txcmpb.2 . . . . . . . . 9 𝑌 = 𝑆
75, 6txuni 21305 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
87ad2antrr 761 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
9 foeq2 6069 . . . . . . 7 ((𝑋 × 𝑌) = (𝑅 ×t 𝑆) → ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑋 ↔ (1st ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑋))
108, 9syl 17 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑋 ↔ (1st ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑋))
114, 10mpbid 222 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑋)
125toptopon 20648 . . . . . . 7 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
136toptopon 20648 . . . . . . 7 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
14 tx1cn 21322 . . . . . . 7 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1512, 13, 14syl2anb 496 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1615ad2antrr 761 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
175cncmp 21105 . . . . 5 (((𝑅 ×t 𝑆) ∈ Comp ∧ (1st ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑋 ∧ (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → 𝑅 ∈ Comp)
181, 11, 16, 17syl3anc 1323 . . . 4 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑅 ∈ Comp)
19 simplrl 799 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑋 ≠ ∅)
20 fo2ndres 7138 . . . . . . 7 (𝑋 ≠ ∅ → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑌)
2119, 20syl 17 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑌)
22 foeq2 6069 . . . . . . 7 ((𝑋 × 𝑌) = (𝑅 ×t 𝑆) → ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑌 ↔ (2nd ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑌))
238, 22syl 17 . . . . . 6 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto𝑌 ↔ (2nd ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑌))
2421, 23mpbid 222 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑌)
25 tx2cn 21323 . . . . . . 7 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
2612, 13, 25syl2anb 496 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
2726ad2antrr 761 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
286cncmp 21105 . . . . 5 (((𝑅 ×t 𝑆) ∈ Comp ∧ (2nd ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑌 ∧ (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → 𝑆 ∈ Comp)
291, 24, 27, 28syl3anc 1323 . . . 4 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑆 ∈ Comp)
3018, 29jca 554 . . 3 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp))
3130ex 450 . 2 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
32 txcmp 21356 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
3331, 32impbid1 215 1 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  c0 3891   cuni 4402   × cxp 5072  cres 5076  ontowfo 5845  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Topctop 20617  TopOnctopon 20618   Cn ccn 20938  Compccmp 21099   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-fin 7903  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-cn 20941  df-cmp 21100  df-tx 21275
This theorem is referenced by: (None)
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