Theorem subrngrng 13964

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 1001	. 2 |- ( ( R e. Rng /\ ( R ↾s A ) e. Rng /\ A C_ ( Base ` R ) ) -> ( R ↾s A ) e. Rng )
2		eqid 2205	. . 3 |- ( Base ` R ) = ( Base ` R )
3	2	issubrng 13961	. 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 2271	. 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 981   = wceq 1373   e. wcel 2176   C_ wss 3166   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833  Rngcrng 13694  SubRngcsubrng 13959

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-fv 5279  df-ov 5947  df-inn 9037  df-ndx 12835  df-slot 12836  df-base 12838  df-subrng 13960

This theorem is referenced by:  subrngsubg  13966  subrngmcl  13971  subsubrng  13976