Theorem txcmpb 23784

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 489	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
2		simplrr 789	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑌 ≠ ∅)
3		fo1stres 8015	. . . . . . 7 ⊢ (𝑌 ≠ ∅ → (1st ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑋)
4	2, 3	syl 18	. . . . . 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 23732	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
8	7	ad2antrr 738	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
9		foeq2 6793	. . . . . . 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 18	. . . . . 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 235	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑋)
12	5	toptopon 23057	. . . . . . 7 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
13	6	toptopon 23057	. . . . . . 7 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
14		tx1cn 23749	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
15	12, 13, 14	syl2anb 609	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
16	15	ad2antrr 738	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
17	5	cncmp 23532	. . . . 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 1396	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑅 ∈ Comp)
19		simplrl 788	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑋 ≠ ∅)
20		fo2ndres 8016	. . . . . . 7 ⊢ (𝑋 ≠ ∅ → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑌)
21	19, 20	syl 18	. . . . . 6 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)):(𝑋 × 𝑌)–onto→𝑌)
22		foeq2 6793	. . . . . . 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 18	. . . . . 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 235	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)):∪ (𝑅 ×t 𝑆)–onto→𝑌)
25		tx2cn 23750	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
26	12, 13, 25	syl2anb 609	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
27	26	ad2antrr 738	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
28	6	cncmp 23532	. . . . 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 1396	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑆 ∈ Comp)
30	18, 29	jca 520	. . 3 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp))
31	30	ex 417	. 2 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
32		txcmp 23783	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
33	31, 32	impbid1 228	1 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp ↔ (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1568   ∈ wcel 2150   ≠ wne 2965  ∅c0 4294  ∪ cuni 4877   × cxp 5663   ↾ cres 5667  –onto→wfo 6538  ‘cfv 6540  (class class class)co 7414  1^st c1st 7987  2^nd c2nd 7988  Topctop 23033  TopOnctopon 23050   Cn ccn 23364  Compccmp 23526   ×_t ctx 23700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-iin 4964  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5560  df-eprel 5565  df-po 5573  df-so 5574  df-fr 5618  df-we 5620  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7866  df-1st 7989  df-2nd 7990  df-1o 8456  df-2o 8457  df-map 8829  df-en 8947  df-dom 8948  df-fin 8950  df-topgen 17499  df-top 23034  df-topon 23051  df-bases 23086  df-cn 23367  df-cmp 23527  df-tx 23702

This theorem is referenced by: (None)