|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > subrngss | Structured version Visualization version GIF version | ||
| Description: A subring is a subset. (Contributed by AV, 14-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| subrngss.1 | ⊢ 𝐵 = (Base‘𝑅) | 
| Ref | Expression | 
|---|---|
| subrngss | ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | subrngss.1 | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 2 | 1 | issubrng 20547 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)) | 
| 3 | 2 | simp3bi 1148 | 1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 Rngcrng 20149 SubRngcsubrng 20545 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-subrng 20546 | 
| This theorem is referenced by: subrngsubg 20552 subrngmre 20562 subsubrng 20563 rhmimasubrnglem 20565 rhmimasubrng 20566 | 
| Copyright terms: Public domain | W3C validator |