Theorem trggrp 23429

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 23428	. 2 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Ring)
2		ringgrp 19883	. 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
3	1, 2	syl 17	1 ⊢ (𝑅 ∈ TopRing → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2105  Grpcgrp 18673  Ringcrg 19878  TopRingctrg 23413

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707  ax-nul 5250

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2941  df-ral 3062  df-rab 3404  df-v 3443  df-sbc 3728  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487  df-ov 7340  df-ring 19880  df-trg 23417

This theorem is referenced by: (None)