Theorem subrngrng 13662

Description: A subring is a non-unital ring. (Contributed by AV, 14-Feb-2025.)

Hypothesis

Ref	Expression
subrngrng.1	|- S = ( R ↾s A )

Assertion

Ref	Expression
subrngrng	|- ( A e. (SubRng ` R ) -> S e. Rng )

Proof of Theorem subrngrng

Step	Hyp	Ref	Expression
1		simp2 1000	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) -> ( R ↾s A ) e. Rng )
2		eqid 2189	. . 3 |- ( Base ` R ) = ( Base ` R )
3	2	issubrng 13659	. 2 |- ( A e. (SubRng ` R ) <-> ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) )
4		subrngrng.1	. . 3 |- S = ( R ↾s A )
5	4	eleq1i 2255	. 2 |- ( S e. Rng <-> ( R ↾s A ) e. Rng )
6	1, 3, 5	3imtr4i 201	1 |- ( A e. (SubRng ` R ) -> S e. Rng )

Syntax hints:   -> wi 4   /\ w3a 980   = wceq 1364   e. wcel 2160   C_ wss 3149   ` cfv 5242  (class class class)co 5906   Basecbs 12592   ↾_s cress 12593  Rngcrng 13392  SubRngcsubrng 13657

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-cnex 7949  ax-resscn 7950  ax-1re 7952  ax-addrcl 7955

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-un 3153  df-in 3155  df-ss 3162  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-fv 5250  df-ov 5909  df-inn 8969  df-ndx 12595  df-slot 12596  df-base 12598  df-subrng 13658

This theorem is referenced by:  subrngsubg  13664  subrngmcl  13669  subsubrng  13674