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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 23988 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd)) |
3 | 2 | simp3bi 1146 | 1 ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ‘cfv 6543 mulGrpcmgp 20035 Ringcrg 20134 TopMndctmd 23894 TopGrpctgp 23895 TopRingctrg 23980 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-trg 23984 |
This theorem is referenced by: mulrcn 24003 cnmpt1mulr 24006 cnmpt2mulr 24007 nrgtdrg 24530 iistmd 33347 |
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