Theorem istrg 23311

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 3908	. . 3 ⊢ (𝑅 ∈ (TopGrp ∩ Ring) ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring))
2	1	anbi1i 624	. 2 ⊢ ((𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
3		fveq2 6769	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4		istrg.1	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅)
5	3, 4	eqtr4di 2798	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6	5	eleq1d 2825	. . 3 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd))
7		df-trg 23307	. . 3 ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
8	6, 7	elrab2 3629	. 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd))
9		df-3an 1088	. 2 ⊢ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd))
10	2, 8, 9	3bitr4i 303	1 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110   ∩ cin 3891  ‘cfv 6431  mulGrpcmgp 19716  Ringcrg 19779  TopMndctmd 23217  TopGrpctgp 23218  TopRingctrg 23303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-iota 6389  df-fv 6439  df-trg 23307

This theorem is referenced by:  trgtmd  23312  trgtgp  23315  trgring  23318  nrgtrg  23850