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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 3963 | . . 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 6897 | . . . . 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 2814 | . . 3 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ∈ TopMnd ↔ 𝑀 ∈ TopMnd)) |
7 | df-trg 24063 | . . 3 ⊢ TopRing = {𝑟 ∈ (TopGrp ∩ Ring) ∣ (mulGrp‘𝑟) ∈ TopMnd} | |
8 | 6, 7 | elrab2 3685 | . 2 ⊢ (𝑅 ∈ TopRing ↔ (𝑅 ∈ (TopGrp ∩ Ring) ∧ 𝑀 ∈ TopMnd)) |
9 | df-3an 1087 | . 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 1085 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 ‘cfv 6548 mulGrpcmgp 20073 Ringcrg 20172 TopMndctmd 23973 TopGrpctgp 23974 TopRingctrg 24059 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-trg 24063 |
This theorem is referenced by: trgtmd 24068 trgtgp 24071 trgring 24074 nrgtrg 24606 |
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