![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subrngrng | Structured version Visualization version GIF version |
Description: A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.) |
Ref | Expression |
---|---|
subrngrng.1 | ⊢ 𝑆 = (𝑅 ↾s 𝐴) |
Ref | Expression |
---|---|
subrngrng | ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1137 | . 2 ⊢ ((𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅)) → (𝑅 ↾s 𝐴) ∈ Rng) | |
2 | eqid 2734 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 2 | issubrng 20568 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ (Base‘𝑅))) |
4 | subrngrng.1 | . . 3 ⊢ 𝑆 = (𝑅 ↾s 𝐴) | |
5 | 4 | eleq1i 2829 | . 2 ⊢ (𝑆 ∈ Rng ↔ (𝑅 ↾s 𝐴) ∈ Rng) |
6 | 1, 3, 5 | 3imtr4i 292 | 1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝑆 ∈ Rng) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1537 ∈ wcel 2103 ⊆ wss 3970 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 ↾s cress 17282 Rngcrng 20174 SubRngcsubrng 20566 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fv 6580 df-ov 7448 df-subrng 20567 |
This theorem is referenced by: subrngsubg 20573 subrngmcl 20578 subsubrng 20584 pzriprnglem7 21516 |
Copyright terms: Public domain | W3C validator |