Theorem trggrp 24133

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 24132	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
2		ringgrp 20190	. 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2114  Grpcgrp 18880  Ringcrg 20185  TopRingctrg 24117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5255

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6458  df-fv 6510  df-ov 7373  df-ring 20187  df-trg 24121

This theorem is referenced by: (None)