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Mirrors > Home > MPE Home > Th. List > trgtgp | Structured version Visualization version GIF version |
Description: A topological ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
trgtgp | ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2823 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | istrg 22774 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ (mulGrp‘𝑅) ∈ TopMnd)) |
3 | 2 | simp1bi 1141 | 1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ‘cfv 6357 mulGrpcmgp 19241 Ringcrg 19299 TopMndctmd 22680 TopGrpctgp 22681 TopRingctrg 22766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-trg 22770 |
This theorem is referenced by: trgtmd2 22779 trgtps 22780 pl1cn 31200 |
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