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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 2825 | . . 3 ⊢ ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) | |
2 | 1 | txval 21745 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)))) |
3 | topbas 21154 | . . . 4 ⊢ (𝑅 ∈ Top → 𝑅 ∈ TopBases) | |
4 | topbas 21154 | . . . 4 ⊢ (𝑆 ∈ Top → 𝑆 ∈ TopBases) | |
5 | 1 | txbas 21748 | . . . 4 ⊢ ((𝑅 ∈ TopBases ∧ 𝑆 ∈ TopBases) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
6 | 3, 4, 5 | syl2an 589 | . . 3 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → ran (𝑢 ∈ 𝑅, 𝑣 ∈ 𝑆 ↦ (𝑢 × 𝑣)) ∈ TopBases) |
7 | tgcl 21151 | . . 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 2906 | 1 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2164 × cxp 5344 ran crn 5347 ‘cfv 6127 (class class class)co 6910 ↦ cmpt2 6912 topGenctg 16458 Topctop 21075 TopBasesctb 21127 ×t ctx 21741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-1st 7433 df-2nd 7434 df-topgen 16464 df-top 21076 df-bases 21128 df-tx 21743 |
This theorem is referenced by: txtopi 21771 txtopon 21772 txcld 21784 neitx 21788 txlly 21817 txnlly 21818 txcmplem1 21822 txcmp 21824 hausdiag 21826 txhaus 21828 tx1stc 21831 txkgen 21833 xkococn 21841 xkoinjcn 21868 txconn 21870 imasnopn 21871 imasncls 21873 utop2nei 22431 utop3cls 22432 qtophaus 30444 txpconn 31756 |
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