Theorem txcmp 23666

Description: The topological product of two compact spaces is compact. (Contributed by Mario Carneiro, 14-Sep-2014.) (Proof shortened 21-Mar-2015.)

Assertion

Ref	Expression
txcmp	⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)

Proof of Theorem txcmp

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cmptop 23418	. . 3 ⊢ (𝑅 ∈ Comp → 𝑅 ∈ Top)
2		cmptop 23418	. . 3 ⊢ (𝑆 ∈ Comp → 𝑆 ∈ Top)
3		txtop 23592	. . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Top)
5		eqid 2734	. . . . . 6 ⊢ ∪ 𝑅 = ∪ 𝑅
6		eqid 2734	. . . . . 6 ⊢ ∪ 𝑆 = ∪ 𝑆
7		simpll 767	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑅 ∈ Comp)
8		simplr 769	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑆 ∈ Comp)
9		elpwi 4611	. . . . . . 7 ⊢ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) → 𝑤 ⊆ (𝑅 ×t 𝑆))
10	9	ad2antrl 728	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → 𝑤 ⊆ (𝑅 ×t 𝑆))
11	5, 6	txuni 23615	. . . . . . . . 9 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
12	1, 2, 11	syl2an 596	. . . . . . . 8 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
13	12	adantr 480	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∪ 𝑅 × ∪ 𝑆) = ∪ (𝑅 ×t 𝑆))
14		simprr 773	. . . . . . 7 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)
15	13, 14	eqtrd 2774	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∪ 𝑅 × ∪ 𝑆) = ∪ 𝑤)
16	5, 6, 7, 8, 10, 15	txcmplem2 23665	. . . . 5 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)(∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣)
17	13	eqeq1d 2736	. . . . . 6 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ((∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣 ↔ ∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
18	17	rexbidv 3176	. . . . 5 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → (∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)(∪ 𝑅 × ∪ 𝑆) = ∪ 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
19	16, 18	mpbid 232	. . . 4 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ (𝑤 ∈ 𝒫 (𝑅 ×t 𝑆) ∧ ∪ (𝑅 ×t 𝑆) = ∪ 𝑤)) → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣)
20	19	expr 456	. . 3 ⊢ (((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) ∧ 𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)) → (∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
21	20	ralrimiva 3143	. 2 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)(∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣))
22		eqid 2734	. . 3 ⊢ ∪ (𝑅 ×t 𝑆) = ∪ (𝑅 ×t 𝑆)
23	22	iscmp 23411	. 2 ⊢ ((𝑅 ×t 𝑆) ∈ Comp ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ ∀𝑤 ∈ 𝒫 (𝑅 ×t 𝑆)(∪ (𝑅 ×t 𝑆) = ∪ 𝑤 → ∃𝑣 ∈ (𝒫 𝑤 ∩ Fin)∪ (𝑅 ×t 𝑆) = ∪ 𝑣)))
24	4, 21, 23	sylanbrc 583	1 ⊢ ((𝑅 ∈ Comp ∧ 𝑆 ∈ Comp) → (𝑅 ×t 𝑆) ∈ Comp)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  ∪ cuni 4911   × cxp 5686  (class class class)co 7430  Fincfn 8983  Topctop 22914  Compccmp 23409   ×_t ctx 23583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-1o 8504  df-2o 8505  df-en 8984  df-dom 8985  df-fin 8987  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-cmp 23410  df-tx 23585

This theorem is referenced by:  txcmpb  23667  txkgen  23675  ptcmpfi  23836  xkohmeo  23838  cnheiborlem  24999