Theorem trggrp 24087

Description: A topological ring is a group. (Contributed by Mario Carneiro, 5-Oct-2015.)

Assertion

Ref	Expression
trggrp	⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Proof of Theorem trggrp

Step	Hyp	Ref	Expression
1		trgring 24086	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
2		ringgrp 20156	. 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2111  Grpcgrp 18846  Ringcrg 20151  TopRingctrg 24071

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 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489  df-ov 7349  df-ring 20153  df-trg 24075

This theorem is referenced by: (None)