Theorem isrngd 14097

Description: Properties that determine a non-unital ring. (Contributed by AV, 14-Feb-2025.)

Hypotheses

Ref	Expression
isrngd.b	|- ( ph -> B = ( Base ` R ) )
isrngd.p	|- ( ph -> .+ = ( +g ` R ) )
isrngd.t	|- ( ph -> .x. = ( .r ` R ) )
isrngd.g	|- ( ph -> R e. Abel )
isrngd.c	|- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B )
isrngd.a	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )
isrngd.d	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) )
isrngd.e	|- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )

Assertion

Ref	Expression
isrngd	|- ( ph -> R e. Rng )

Distinct variable groups:   x, y, z, B   ph, x, y, z   x, R, y, z

Allowed substitution hints:   .+ ( x, y, z)   .x. ( x, y, z)

Proof of Theorem isrngd

Step	Hyp	Ref	Expression
1		isrngd.g	. 2 |- ( ph -> R e. Abel )
2		isrngd.b	. . . 4 |- ( ph -> B = ( Base ` R ) )
3		eqid 2232	. . . . . 6 |- (mulGrp ` R ) = (mulGrp ` R )
4		eqid 2232	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
5	3, 4	mgpbasg 14070	. . . . 5 |- ( R e. Abel -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
6	1, 5	syl 14	. . . 4 |- ( ph -> ( Base ` R ) = ( Base ` (mulGrp ` R ) ) )
7	2, 6	eqtrd 2265	. . 3 |- ( ph -> B = ( Base ` (mulGrp ` R ) ) )
8		isrngd.t	. . . 4 |- ( ph -> .x. = ( .r ` R ) )
9		eqid 2232	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
10	3, 9	mgpplusgg 14068	. . . . 5 |- ( R e. Abel -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
11	1, 10	syl 14	. . . 4 |- ( ph -> ( .r ` R ) = ( +g ` (mulGrp ` R ) ) )
12	8, 11	eqtrd 2265	. . 3 |- ( ph -> .x. = ( +g ` (mulGrp ` R ) ) )
13		isrngd.c	. . 3 |- ( ( ph /\ x e. B /\ y e. B ) -> ( x .x. y ) e. B )
14		isrngd.a	. . 3 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .x. z ) = ( x .x. ( y .x. z ) ) )
15	3	mgpex 14069	. . . 4 |- ( R e. Abel -> (mulGrp ` R ) e. _V )
16	1, 15	syl 14	. . 3 |- ( ph -> (mulGrp ` R ) e. _V )
17	7, 12, 13, 14, 16	issgrpd 13625	. 2 |- ( ph -> (mulGrp ` R ) e. Smgrp )
18	2	eleq2d 2302	. . . . . 6 |- ( ph -> ( x e. B <-> x e. ( Base ` R ) ) )
19	2	eleq2d 2302	. . . . . 6 |- ( ph -> ( y e. B <-> y e. ( Base ` R ) ) )
20	2	eleq2d 2302	. . . . . 6 |- ( ph -> ( z e. B <-> z e. ( Base ` R ) ) )
21	18, 19, 20	3anbi123d 1349	. . . . 5 |- ( ph -> ( ( x e. B /\ y e. B /\ z e. B ) <-> ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) )
22	21	biimpar 297	. . . 4 |- ( ( ph /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( x e. B /\ y e. B /\ z e. B ) )
23		isrngd.d	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( ( x .x. y ) .+ ( x .x. z ) ) )
24	8	adantr 276	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> .x. = ( .r ` R ) )
25		eqidd 2233	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> x = x )
26		isrngd.p	. . . . . . . 8 |- ( ph -> .+ = ( +g ` R ) )
27	26	oveqdr 6078	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( y .+ z ) = ( y ( +g ` R ) z ) )
28	24, 25, 27	oveq123d 6071	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. ( y .+ z ) ) = ( x ( .r ` R ) ( y ( +g ` R ) z ) ) )
29	26	adantr 276	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> .+ = ( +g ` R ) )
30	8	oveqdr 6078	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. y ) = ( x ( .r ` R ) y ) )
31	8	oveqdr 6078	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .x. z ) = ( x ( .r ` R ) z ) )
32	29, 30, 31	oveq123d 6071	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. y ) .+ ( x .x. z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) )
33	23, 28, 32	3eqtr3d 2273	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) )
34		isrngd.e	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x .x. z ) .+ ( y .x. z ) ) )
35	26	oveqdr 6078	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( x .+ y ) = ( x ( +g ` R ) y ) )
36		eqidd 2233	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> z = z )
37	24, 35, 36	oveq123d 6071	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .+ y ) .x. z ) = ( ( x ( +g ` R ) y ) ( .r ` R ) z ) )
38	8	oveqdr 6078	. . . . . . 7 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( y .x. z ) = ( y ( .r ` R ) z ) )
39	29, 31, 38	oveq123d 6071	. . . . . 6 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x .x. z ) .+ ( y .x. z ) ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) )
40	34, 37, 39	3eqtr3d 2273	. . . . 5 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) )
41	33, 40	jca 306	. . . 4 |- ( ( ph /\ ( x e. B /\ y e. B /\ z e. B ) ) -> ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) )
42	22, 41	syldan 282	. . 3 |- ( ( ph /\ ( x e. ( Base ` R ) /\ y e. ( Base ` R ) /\ z e. ( Base ` R ) ) ) -> ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) )
43	42	ralrimivvva 2625	. 2 |- ( ph -> A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) )
44		eqid 2232	. . 3 |- ( +g ` R ) = ( +g ` R )
45	4, 3, 44, 9	isrng 14078	. 2 |- ( R e. Rng <-> ( R e. Abel /\ (mulGrp ` R ) e. Smgrp /\ A. x e. ( Base ` R ) A. y e. ( Base ` R ) A. z e. ( Base ` R ) ( ( x ( .r ` R ) ( y ( +g ` R ) z ) ) = ( ( x ( .r ` R ) y ) ( +g ` R ) ( x ( .r ` R ) z ) ) /\ ( ( x ( +g ` R ) y ) ( .r ` R ) z ) = ( ( x ( .r ` R ) z ) ( +g ` R ) ( y ( .r ` R ) z ) ) ) ) )
46	1, 17, 43, 45	syl3anbrc 1208	1 |- ( ph -> R e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   ` cfv 5352  (class class class)co 6050   Basecbs 13212   +g cplusg 13290   .rcmulr 13291  Smgrpcsgrp 13614   Abelcabl 14002  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:  rngressid  14098  imasrng  14100  opprrng  14221  issubrng2  14355