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Mirrors > Home > MPE Home > Th. List > istrg | Structured version Visualization version GIF version |
Description: Express the predicate "𝑅 is a topological ring". (Contributed by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
istrg.1 | ⊢ 𝑀 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
istrg | ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring ∧ 𝑀 ∈ TopMnd)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elin 3957 | . . 3 ⊢ (𝑅 ∈ (TopGrp ∩ Ring) ↔ (𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring)) | |
2 | 1 | anbi1i 623 | . 2 ⊢ ((𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd) ↔ ((𝑅 ∈ TopGrp ∧ 𝑅 ∈ Ring) ∧ 𝑀 ∈ TopMnd)) |
3 | fveq2 6882 | . . . . 5 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
4 | istrg.1 | . . . . 5 ⊢ 𝑀 = (mulGrp‘𝑅) | |
5 | 3, 4 | eqtr4di 2782 | . . . 4 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = 𝑀) |
6 | 5 | eleq1d 2810 | . . 3 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd)) |
7 | df-trg 24008 | . . 3 ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd} | |
8 | 6, 7 | elrab2 3679 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd)) |
9 | df-3an 1086 | . 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)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∩ cin 3940 ‘cfv 6534 mulGrpcmgp 20035 Ringcrg 20134 TopMndctmd 23918 TopGrpctgp 23919 TopRingctrg 24004 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-iota 6486 df-fv 6542 df-trg 24008 |
This theorem is referenced by: trgtmd 24013 trgtgp 24016 trgring 24019 nrgtrg 24551 |
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