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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 22344 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd)) |
3 | 2 | simp3bi 1181 | 1 ⊢ (𝑅 ∈ TopRing → 𝑀 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 mulGrpcmgp 18850 Ringcrg 18908 TopMndctmd 22251 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 |
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-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rex 3123 df-rab 3126 df-v 3416 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-trg 22340 |
This theorem is referenced by: mulrcn 22359 cnmpt1mulr 22362 cnmpt2mulr 22363 nrgtdrg 22874 iistmd 30489 |
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