Theorem istrg 24099

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 3914	. . 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 6831	. . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅))
4		istrg.1	. . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅)
5	3, 4	eqtr4di 2786	. . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀)
6	5	eleq1d 2818	. . 3 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd))
7		df-trg 24095	. . 3 ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd}
8	6, 7	elrab2 3646	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113   ∩ cin 3897  ‘cfv 6489  mulGrpcmgp 20066  Ringcrg 20159  TopMndctmd 24005  TopGrpctgp 24006  TopRingctrg 24091

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 2115  ax-9 2123  ax-ext 2705

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 2712  df-cleq 2725  df-clel 2808  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-trg 24095

This theorem is referenced by:  trgtmd  24100  trgtgp  24103  trgring  24106  nrgtrg  24625