Theorem txcmpb 22795

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

Step	Hyp	Ref	Expression
1		simpr 485	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
2		simplrr 775	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑌 ≠ ∅)
3		fo1stres 7857	. . . . . . 7 ⊢ (𝑌 ≠ ∅ → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑋)
4	2, 3	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑋)
5		txcmpb.1	. . . . . . . . 9 ⊢ 𝑋 = ∪ 𝑅
6		txcmpb.2	. . . . . . . . 9 ⊢ 𝑌 = ∪ 𝑆
7	5, 6	txuni 22743	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
8	7	ad2antrr 723	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
9		foeq2 6685	. . . . . . 7 ⊢ ((𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) → ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑋 ↔ (1st ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑋))
10	8, 9	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → ((1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑋 ↔ (1st ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑋))
11	4, 10	mpbid 231	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑋)
12	5	toptopon 22066	. . . . . . 7 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
13	6	toptopon 22066	. . . . . . 7 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
14		tx1cn 22760	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
15	12, 13, 14	syl2anb 598	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
16	15	ad2antrr 723	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
17	5	cncmp 22543	. . . . 5 ⊢ (((𝑅 ×t 𝑆) ∈ Comp ∧ (1st ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑋 ∧ (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅)) → 𝑅 ∈ Comp)
18	1, 11, 16, 17	syl3anc 1370	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑅 ∈ Comp)
19		simplrl 774	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑋 ≠ ∅)
20		fo2ndres 7858	. . . . . . 7 ⊢ (𝑋 ≠ ∅ → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑌)
21	19, 20	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑌)
22		foeq2 6685	. . . . . . 7 ⊢ ((𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆) → ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑌 ↔ (2nd ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑌))
23	8, 22	syl 17	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → ((2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑌 ↔ (2nd ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑌))
24	21, 23	mpbid 231	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑌)
25		tx2cn 22761	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
26	12, 13, 25	syl2anb 598	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
27	26	ad2antrr 723	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
28	6	cncmp 22543	. . . . 5 ⊢ (((𝑅 ×t 𝑆) ∈ Comp ∧ (2nd ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑌 ∧ (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆)) → 𝑆 ∈ Comp)
29	1, 24, 27, 28	syl3anc 1370	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑆 ∈ Comp)
30	18, 29	jca 512	. . 3 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp))
31	30	ex 413	. 2 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
32		txcmp 22794	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
33	31, 32	impbid1 224	1 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ≠ wne 2943  ∅c0 4256  ∪ cuni 4839   × cxp 5587   ↾ cres 5591  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  Topctop 22042  TopOnctopon 22059   Cn ccn 22375  Compccmp 22537   ×_t ctx 22711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-fin 8737  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-cn 22378  df-cmp 22538  df-tx 22713

This theorem is referenced by: (None)