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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 20806 | . 2 ⊢ (𝑆 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝑆 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾s 𝑆) ∈ DivRing)) | |
2 | sdrgid.1 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
3 | 2 | subrgss 20595 | . . 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 2103 ⊆ wss 3970 ‘cfv 6572 (class class class)co 7445 Basecbs 17253 ↾s cress 17282 SubRingcsubrg 20590 DivRingcdr 20746 SubDRingcsdrg 20804 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 ax-sep 5320 ax-nul 5327 ax-pow 5386 ax-pr 5450 |
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 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3439 df-v 3484 df-dif 3973 df-un 3975 df-in 3977 df-ss 3987 df-nul 4348 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5170 df-opab 5232 df-mpt 5253 df-id 5597 df-xp 5705 df-rel 5706 df-cnv 5707 df-co 5708 df-dm 5709 df-rn 5710 df-res 5711 df-ima 5712 df-iota 6524 df-fun 6574 df-fv 6580 df-ov 7448 df-subrg 20592 df-sdrg 20805 |
This theorem is referenced by: sdrgbas 20812 fldgenidfld 33276 fldgenfldext 33670 evls1fldgencl 33672 algextdeglem8 33707 rtelextdg2lem 33709 rtelextdg2 33710 constrelextdg2 33729 |
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