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Mirrors > Home > MPE Home > Th. List > trgtps | Structured version Visualization version GIF version |
Description: A topological ring is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
trgtps | ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trgtgp 22348 | . 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) | |
2 | tgptps 22261 | . 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopSp) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopSp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 TopSpctps 21114 TopGrpctgp 22252 TopRingctrg 22336 |
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-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-nul 5015 |
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-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-iota 6090 df-fv 6135 df-ov 6913 df-tmd 22253 df-tgp 22254 df-trg 22340 |
This theorem is referenced by: tdrgtps 22357 tlmscatps 22371 |
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