Theorem rngcl 14088

Description: Closure of the multiplication operation of a non-unital ring. (Contributed by AV, 17-Apr-2020.)

Hypotheses

Ref	Expression
rngcl.b	|- B = ( Base ` R )
rngcl.t	|- .x. = ( .r ` R )

Assertion

Ref	Expression
rngcl	|- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )

Proof of Theorem rngcl

Step	Hyp	Ref	Expression
1		eqid 2232	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
2	1	rngmgp 14080	. . . . 5 |- ( R e. Rng -> (mulGrp ` R ) e. Smgrp )
3		sgrpmgm 13620	. . . . 5 |- ( (mulGrp ` R ) e. Smgrp -> (mulGrp ` R ) e. Mgm )
4	2, 3	syl 14	. . . 4 |- ( R e. Rng -> (mulGrp ` R ) e. Mgm )
5	4	3ad2ant1 1045	. . 3 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> (mulGrp ` R ) e. Mgm )
6		simp2 1025	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> X e. B )
7		rngcl.b	. . . . . 6 |- B = ( Base ` R )
8	1, 7	mgpbasg 14070	. . . . 5 |- ( R e. Rng -> B = ( Base ` (mulGrp ` R ) ) )
9	8	3ad2ant1 1045	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> B = ( Base ` (mulGrp ` R ) ) )
10	6, 9	eleqtrd 2311	. . 3 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> X e. ( Base ` (mulGrp ` R ) ) )
11		simp3 1026	. . . 4 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> Y e. B )
12	11, 9	eleqtrd 2311	. . 3 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> Y e. ( Base ` (mulGrp ` R ) ) )
13		eqid 2232	. . . 4 |- ( Base ` (mulGrp ` R ) ) = ( Base ` (mulGrp ` R ) )
14		eqid 2232	. . . 4 |- ( +g ` (mulGrp ` R ) ) = ( +g ` (mulGrp ` R ) )
15	13, 14	mgmcl 13572	. . 3 |- ( ( (mulGrp ` R ) e. Mgm /\ X e. ( Base ` (mulGrp ` R ) ) /\ Y e. ( Base ` (mulGrp ` R ) ) ) -> ( X ( +g ` (mulGrp ` R ) ) Y ) e. ( Base ` (mulGrp ` R ) ) )
16	5, 10, 12, 15	syl3anc 1274	. 2 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X ( +g ` (mulGrp ` R ) ) Y ) e. ( Base ` (mulGrp ` R ) ) )
17		rngcl.t	. . . . 5 |- .x. = ( .r ` R )
18	1, 17	mgpplusgg 14068	. . . 4 |- ( R e. Rng -> .x. = ( +g ` (mulGrp ` R ) ) )
19	18	oveqd 6067	. . 3 |- ( R e. Rng -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
20	19	3ad2ant1 1045	. 2 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) = ( X ( +g ` (mulGrp ` R ) ) Y ) )
21	16, 20, 9	3eltr4d 2316	1 |- ( ( R e. Rng /\ X e. B /\ Y e. B ) -> ( X .x. Y ) e. B )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291  Mgmcmgm 13567  Smgrpcsgrp 13614  mulGrpcmgp 14064  Rngcrng 14076

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-pre-ltirr 8239  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-pnf 8310  df-mnf 8311  df-ltxr 8313  df-inn 9238  df-2 9296  df-3 9297  df-ndx 13215  df-slot 13216  df-base 13218  df-sets 13219  df-plusg 13303  df-mulr 13304  df-mgm 13569  df-sgrp 13615  df-mgp 14065  df-rng 14077

This theorem is referenced by:  rnglz  14089  rngrz  14090  rngmneg1  14091  rngmneg2  14092  rngm2neg  14093  rngsubdi  14095  rngsubdir  14096  rngressid  14098  imasrng  14100  qusrng  14102  opprrng  14221  subrngmcl  14354  rnglidlmcl  14628  2idlcpblrng  14671  qusmulrng  14680