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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 20815 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | |
2 | sdrgid.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 2 | subrgss 20598 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
4 | 3 | 3ad2ant2 1135 | . 2 ⊢ ((𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing) → 𝑆 ⊆ 𝐵) |
5 | 1, 4 | sylbi 217 | 1 ⊢ (𝑆 ∈ (SubDRing‘𝑅) → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1539 ∈ wcel 2108 ⊆ wss 3966 ‘cfv 6569 (class class class)co 7438 Basecbs 17254 ↾s cress 17283 SubRingcsubrg 20595 DivRingcdr 20755 SubDRingcsdrg 20813 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 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 5305 ax-nul 5315 ax-pow 5374 ax-pr 5441 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 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 3483 df-dif 3969 df-un 3971 df-in 3973 df-ss 3983 df-nul 4343 df-if 4535 df-pw 4610 df-sn 4635 df-pr 4637 df-op 4641 df-uni 4916 df-br 5152 df-opab 5214 df-mpt 5235 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-res 5705 df-ima 5706 df-iota 6522 df-fun 6571 df-fv 6577 df-ov 7441 df-subrg 20596 df-sdrg 20814 |
This theorem is referenced by: sdrgbas 20821 fldgenidfld 33331 fldgenfldext 33725 evls1fldgencl 33727 algextdeglem8 33762 rtelextdg2lem 33764 rtelextdg2 33765 constrelextdg2 33784 |
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