MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trgtmd2 Structured version   Visualization version   GIF version

Theorem trgtmd2 22380
Description: A topological ring is a topological monoid. (Contributed by Mario Carneiro, 5-Oct-2015.)
Assertion
Ref Expression
trgtmd2 (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)

Proof of Theorem trgtmd2
StepHypRef Expression
1 trgtgp 22379 . 2 (𝑅 ∈ TopRing → 𝑅 ∈ TopGrp)
2 tgptmd 22291 . 2 (𝑅 ∈ TopGrp → 𝑅 ∈ TopMnd)
31, 2syl 17 1 (𝑅 ∈ TopRing → 𝑅 ∈ TopMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  TopMndctmd 22282  TopGrpctgp 22283  TopRingctrg 22367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-tgp 22285  df-trg 22371
This theorem is referenced by:  tdrgtmd  22387  qqhcn  30633
  Copyright terms: Public domain W3C validator