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Mirrors > Home > MPE Home > Th. List > txtopi | Structured version Visualization version GIF version |
Description: The product of two topologies is a topology. (Contributed by Jeff Madsen, 15-Jun-2010.) |
Ref | Expression |
---|---|
txtopi.1 | ⊢ 𝑅 ∈ Top |
txtopi.2 | ⊢ 𝑆 ∈ Top |
Ref | Expression |
---|---|
txtopi | ⊢ (𝑅 ×t 𝑆) ∈ Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | txtopi.1 | . 2 ⊢ 𝑅 ∈ Top | |
2 | txtopi.2 | . 2 ⊢ 𝑆 ∈ Top | |
3 | txtop 22716 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) | |
4 | 1, 2, 3 | mp2an 689 | 1 ⊢ (𝑅 ×t 𝑆) ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2110 (class class class)co 7269 Topctop 22038 ×t ctx 22707 |
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 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-fv 6439 df-ov 7272 df-oprab 7273 df-mpo 7274 df-1st 7822 df-2nd 7823 df-topgen 17150 df-top 22039 df-bases 22092 df-tx 22709 |
This theorem is referenced by: sxbrsigalem3 32233 dya2iocucvr 32245 cvmlift2lem9 33267 cvmlift2lem11 33269 cvmlift2lem12 33270 |
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