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| 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 24177 | . 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) | |
| 2 | tgptps 24089 | . 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 2107 TopSpctps 22939 TopGrpctgp 24080 TopRingctrg 24165 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 df-tmd 24081 df-tgp 24082 df-trg 24169 | 
| This theorem is referenced by: tdrgtps 24186 tlmscatps 24200 | 
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