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Mirrors > Home > MPE Home > Th. List > trgtmd | Structured version Visualization version GIF version |
Description: The multiplicative monoid of a topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istrg.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
trgtmd | ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istrg.1 | . . 3 ⊢ 𝑀 = (mulGrp‘𝑅) | |
2 | 1 | istrg 23015 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd)) |
3 | 2 | simp3bi 1149 | 1 ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2112 ‘cfv 6358 mulGrpcmgp 19458 Ringcrg 19516 TopMndctmd 22921 TopGrpctgp 22922 TopRingctrg 23007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-trg 23011 |
This theorem is referenced by: mulrcn 23030 cnmpt1mulr 23033 cnmpt2mulr 23034 nrgtdrg 23545 iistmd 31520 |
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