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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 20443 | . 2 ⊢ (𝐴 ∈ (SubRng‘𝑅) ↔ (𝑅 ∈ Rng ∧ (𝑅 ↾s 𝐴) ∈ Rng ∧ 𝐴 ⊆ 𝐵)) |
3 | 2 | simp3bi 1146 | 1 ⊢ (𝐴 ∈ (SubRng‘𝑅) → 𝐴 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 ⊆ wss 3948 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 ↾s cress 17180 Rngcrng 20053 SubRngcsubrng 20441 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-subrng 20442 |
This theorem is referenced by: subrngsubg 20448 subrngmre 20458 subsubrng 20459 rhmimasubrnglem 20461 rhmimasubrng 20462 |
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