sdrgsubrg - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > sdrgsubrg		Structured version Visualization version GIF version

Theorem sdrgsubrg 20756

Description: A sub-division-ring is a subring. (Contributed by SN, 19-Feb-2025.)

**Assertion**
Ref	Expression
sdrgsubrg	⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))

**Proof of Theorem sdrgsubrg**
Step	Hyp	Ref	Expression
1		issdrg 20753	. 2 ⊢ (𝐴 ∈ (SubDRing‘𝑅) ↔ (𝑅 ∈ DivRing ∧ 𝐴 ∈ (SubRing‘𝑅) ∧ (𝑅 ↾_s 𝐴) ∈ DivRing))
2	1	simp2bi 1146	1 ⊢ (𝐴 ∈ (SubDRing‘𝑅) → 𝐴 ∈ (SubRing‘𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∈ wcel 2109 ‘cfv 6536 (class class class)co 7410 ↾_s cress 17256 SubRingcsubrg 20534 DivRingcdr 20694 SubDRingcsdrg 20751

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pr 5407

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rab 3421 df-v 3466 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fv 6544 df-ov 7413 df-sdrg 20752