Theorem txopn 14770

Description: The product of two open sets is open in the product topology. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
txopn	⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))

Proof of Theorem txopn

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

Step	Hyp	Ref	Expression
1		eqid 2205	. . . . . 6 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
2	1	txbasex 14762	. . . . 5 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
3		bastg 14566	. . . . 5 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ V → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
4	2, 3	syl 14	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
5	4	adantr 276	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ⊆ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
6		eqid 2205	. . . . . 6 ⊢ (𝐴 × 𝐵) = (𝐴 × 𝐵)
7		xpeq1 4690	. . . . . . . 8 ⊢ (𝑢 = 𝐴 → (𝑢 × 𝑣) = (𝐴 × 𝑣))
8	7	eqeq2d 2217	. . . . . . 7 ⊢ (𝑢 = 𝐴 → ((𝐴 × 𝐵) = (𝑢 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝑣)))
9		xpeq2 4691	. . . . . . . 8 ⊢ (𝑣 = 𝐵 → (𝐴 × 𝑣) = (𝐴 × 𝐵))
10	9	eqeq2d 2217	. . . . . . 7 ⊢ (𝑣 = 𝐵 → ((𝐴 × 𝐵) = (𝐴 × 𝑣) ↔ (𝐴 × 𝐵) = (𝐴 × 𝐵)))
11	8, 10	rspc2ev 2892	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆 ∧ (𝐴 × 𝐵) = (𝐴 × 𝐵)) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
12	6, 11	mp3an3 1339	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣))
13		xpexg 4790	. . . . . 6 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴 × 𝐵) ∈ V)
14		eqid 2205	. . . . . . 7 ⊢ (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))
15	14	elrnmpog 6060	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ V → ((𝐴 × 𝐵) ∈ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ↔ ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣)))
16	13, 15	syl 14	. . . . 5 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → ((𝐴 × 𝐵) ∈ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ↔ ∃𝑢 ∈ 𝑅 ∃𝑣 ∈ 𝑆 (𝐴 × 𝐵) = (𝑢 × 𝑣)))
17	12, 16	mpbird 167	. . . 4 ⊢ ((𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆) → (𝐴 × 𝐵) ∈ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
18	17	adantl 277	. . 3 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))
19	5, 18	sseldd 3194	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
20	1	txval 14760	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
21	20	adantr 276	. 2 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))))
22	19, 21	eleqtrrd 2285	1 ⊢ (((𝑅 ∈ 𝑉 ∧ 𝑆 ∈ 𝑊) ∧ (𝐴 ∈ 𝑅 ∧ 𝐵 ∈ 𝑆)) → (𝐴 × 𝐵) ∈ (𝑅 ×t 𝑆))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2176  ∃wrex 2485  Vcvv 2772   ⊆ wss 3166   × cxp 4674  ran crn 4677  ‘cfv 5272  (class class class)co 5946   ∈ cmpo 5948  topGenctg 13119   ×_t ctx 14757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-ov 5949  df-oprab 5950  df-mpo 5951  df-1st 6228  df-2nd 6229  df-topgen 13125  df-tx 14758

This theorem is referenced by:  txbasval  14772  neitx  14773  tx1cn  14774  tx2cn  14775  txlm  14784