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Mirrors > Home > MPE Home > Th. List > sdrgss | Structured version Visualization version GIF version |
Description: A division subring is a subset of the base set. (Contributed by Thierry Arnoux, 21-Aug-2023.) |
Ref | Expression |
---|---|
sdrgid.1 | ⊢ 𝐵 = (Base‘𝑅) |
Ref | Expression |
---|---|
sdrgss | ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issdrg 19567 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | |
2 | sdrgid.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 2 | subrgss 19529 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
4 | 3 | 3ad2ant2 1131 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) → 𝑆 ⊆ 𝐵) |
5 | 1, 4 | sylbi 220 | 1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 DivRingcdr 19495 SubRingcsubrg 19524 SubDRingcsdrg 19565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-subrg 19526 df-sdrg 19566 |
This theorem is referenced by: (None) |
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