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Mirrors > Home > MPE Home > Th. List > trgtmd2 | Structured version Visualization version GIF version |
Description: A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
trgtmd2 | ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trgtgp 22770 | . 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp) | |
2 | tgptmd 22681 | . 2 ⊢ (𝑅 ∈ TopGrp → 𝑅 ∈ TopMnd) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 TopMndctmd 22672 TopGrpctgp 22673 TopRingctrg 22758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5202 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-tgp 22675 df-trg 22762 |
This theorem is referenced by: tdrgtmd 22778 qqhcn 31227 |
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