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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 23593 | . 2 ⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top) | |
4 | 1, 2, 3 | mp2an 692 | 1 ⊢ (𝑅 ×t 𝑆) ∈ Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 (class class class)co 7431 Topctop 22915 ×t ctx 23584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-topgen 17490 df-top 22916 df-bases 22969 df-tx 23586 |
This theorem is referenced by: sxbrsigalem3 34254 dya2iocucvr 34266 cvmlift2lem9 35296 cvmlift2lem11 35298 cvmlift2lem12 35299 |
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