Theorem txcmpb 23910

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 490	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)
2		simplrr 790	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑌 ≠ ∅)
3		fo1stres 8011	. . . . . . 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 23858	. . . . . . . 8 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
8	7	ad2antrr 739	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑋 × 𝑌) = ∪ (𝑅 ×t 𝑆))
9		foeq2 6782	. . . . . . 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 23182	. . . . . . 7 ⊢ (𝑅 ∈ Top ↔ 𝑅 ∈ (TopOn‘𝑋))
13	6	toptopon 23182	. . . . . . 7 ⊢ (𝑆 ∈ Top ↔ 𝑆 ∈ (TopOn‘𝑌))
14		tx1cn 23875	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
15	12, 13, 14	syl2anb 610	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
16	15	ad2antrr 739	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (1st ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑅))
17	5	cncmp 23657	. . . . 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 1398	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑅 ∈ Comp)
19		simplrl 789	. . . . . . 7 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑋 ≠ ∅)
20		fo2ndres 8012	. . . . . . 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 6782	. . . . . . 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 23876	. . . . . . 7 ⊢ ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
26	12, 13, 25	syl2anb 610	. . . . . 6 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
27	26	ad2antrr 739	. . . . 5 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (2nd ↾ (𝑋 × 𝑌)) ∈ ((𝑅 ×t 𝑆) Cn 𝑆))
28	6	cncmp 23657	. . . . 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 1398	. . . 4 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → 𝑆 ∈ Comp)
30	18, 29	jca 521	. . 3 ⊢ ((((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) ∧ (𝑅 ×t 𝑆) ∈ Comp) → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp))
31	30	ex 418	. 2 ⊢ (((𝑅 ∈ Top ∧ 𝑆 ∈ Top) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → ((𝑅 ×t 𝑆) ∈ Comp → (𝑅 ∈ Comp ∧ 𝑆 ∈ Comp)))
32		txcmp 23909	. 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∅c0 4279  ∪ cuni 4867   × cxp 5646   ↾ cres 5650  –onto→wfo 6526  ‘cfv 6528  (class class class)co 7409  1^st c1st 7983  2^nd c2nd 7984  Topctop 23158  TopOnctopon 23175   Cn ccn 23489  Compccmp 23651   ×_t ctx 23826

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7862  df-1st 7985  df-2nd 7986  df-1o 8455  df-2o 8456  df-map 8828  df-en 8953  df-dom 8954  df-fin 8956  df-topgen 17561  df-top 23159  df-topon 23176  df-bases 23211  df-cn 23492  df-cmp 23652  df-tx 23828

This theorem is used by: (None)