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Theorem txcmpb 22168
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 485 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
2 simplrr 774 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑌 ≠ ∅)
3 fo1stres 7706 . . . . . . 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 22116 . . . . . . . 8 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
87ad2antrr 722 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
9 foeq2 6584 . . . . . . 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 233 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑋)
125toptopon 21441 . . . . . . 7 (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
136toptopon 21441 . . . . . . 7 (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
14 tx1cn 22133 . . . . . . 7 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1512, 13, 14syl2anb 597 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
1615ad2antrr 722 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
175cncmp 21916 . . . . 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 1365 . . . 4 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑅 ∈ Comp)
19 simplrl 773 . . . . . . 7 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑋 ≠ ∅)
20 fo2ndres 7707 . . . . . . 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 6584 . . . . . . 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 233 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)): (𝑅 ×t 𝑆)–onto𝑌)
25 tx2cn 22134 . . . . . . 7 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
2612, 13, 25syl2anb 597 . . . . . 6 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
2726ad2antrr 722 . . . . 5 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
286cncmp 21916 . . . . 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 1365 . . . 4 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑆 ∈ Comp)
3018, 29jca 512 . . 3 ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp))
3130ex 413 . 2 (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
32 txcmp 22167 . 2 ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
3331, 32impbid1 226 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 207  wa 396   = wceq 1530  wcel 2107  wne 3021  c0 4295   cuni 4837   × cxp 5552  cres 5556  ontowfo 6350  cfv 6352  (class class class)co 7148  1st c1st 7678  2nd c2nd 7679  Topctop 21417  TopOnctopon 21434   Cn ccn 21748  Compccmp 21910   ×t ctx 22084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-iin 4920  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-fin 8502  df-topgen 16707  df-top 21418  df-topon 21435  df-bases 21470  df-cn 21751  df-cmp 21911  df-tx 22086
This theorem is referenced by: (None)
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