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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 20813 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | |
2 | sdrgid.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 2 | subrgss 20602 | . . 3 ⊢ (𝑆 ∈ (SubRing‘𝑅) → 𝑆 ⊆ 𝐵) |
4 | 3 | 3ad2ant2 1134 | . 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 1537 ∈ wcel 2108 ⊆ wss 3976 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 ↾s cress 17289 SubRingcsubrg 20597 DivRingcdr 20753 SubDRingcsdrg 20811 |
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 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 |
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 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6527 df-fun 6577 df-fv 6583 df-ov 7453 df-subrg 20599 df-sdrg 20812 |
This theorem is referenced by: sdrgbas 20819 fldgenidfld 33286 fldgenfldext 33680 evls1fldgencl 33682 algextdeglem8 33717 rtelextdg2lem 33719 rtelextdg2 33720 constrelextdg2 33739 |
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