Theorem istrg 24538

Description: Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypothesis

Ref	Expression
istrg.1	⊢ 𝑀 = (mulGrp‘𝑅)

Assertion

Ref	Expression
istrg	⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))

Proof of Theorem istrg

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 4096	. . 3 ⊢ (𝑅 ∈ (TopGrp ∩ Ring) ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring))
2	1	anbi1i 636	. 2 ⊢ ((𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
3		fveq2 6916	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4		istrg.1	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅)
5	3, 4	eqtr4di 2816	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6	5	eleq1d 2849	. . 3 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd))
7		df-trg 24534	. . 3 ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
8	6, 7	elrab2 3652	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd))
9		df-3an 1105	. 2 ⊢ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
10	2, 8, 9	3bitr4i 306	1 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ∩ cin 4092  ‘cfv 6566  mulGrpcmgp 20411  Ringcrg 20510  TopMndctmd 24444  TopGrpctgp 24445  TopRingctrg 24530

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-iota 6520  df-fv 6574  df-trg 24534

This theorem is used by:  trgtmd  24539  trgtgp  24542  trgring  24545  nrgtrg  25064