Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > txtop | Structured version Visualization version GIF version |
Description: The product of two topologies is a topology. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
txtop | ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
2 | 1 | txval 22308 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
3 | topbas 21716 | . . . 4 ⊢ (𝑅 ∈ Top → 𝑅 ∈ TopBases) | |
4 | topbas 21716 | . . . 4 ⊢ (𝑆 ∈ Top → 𝑆 ∈ TopBases) | |
5 | 1 | txbas 22311 | . . . 4 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
6 | 3, 4, 5 | syl2an 599 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
7 | tgcl 21713 | . . 3 ⊢ (ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) ∈ Top) | |
8 | 6, 7 | syl 17 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣))) ∈ Top) |
9 | 2, 8 | eqeltrd 2833 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2113 × cxp 5517 ran crn 5520 ‘cfv 6333 (class class class)co 7164 ∈ cmpo 7166 topGenctg 16807 Topctop 21637 TopBasesctb 21689 ×t ctx 22304 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-topgen 16813 df-top 21638 df-bases 21690 df-tx 22306 |
This theorem is referenced by: txtopi 22334 txtopon 22335 txcld 22347 neitx 22351 txlly 22380 txnlly 22381 txcmplem1 22385 txcmp 22387 hausdiag 22389 txhaus 22391 tx1stc 22394 txkgen 22396 xkococn 22404 xkoinjcn 22431 txconn 22433 imasnopn 22434 imasncls 22436 utop2nei 22995 utop3cls 22996 qtophaus 31350 txpconn 32757 |
Copyright terms: Public domain | W3C validator |