Theorem rnglidlrng 14260

Description: A (left) ideal of a non-unital ring is a non-unital ring. (Contributed by AV, 17-Feb-2020.) Generalization for non-unital rings. The assumption U e. (SubGrp ` R ) is required because a left ideal of a non-unital ring does not have to be a subgroup. (Revised by AV, 11-Mar-2025.)

Hypotheses

Ref	Expression
rnglidlabl.l	|- L = (LIdeal ` R )
rnglidlabl.i	|- I = ( R ↾s U )

Assertion

Ref	Expression
rnglidlrng	|- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Proof of Theorem rnglidlrng

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngabl 13697	. . . 4 |- ( R e. Rng -> R e. Abel )
2	1	3ad2ant1 1021	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Abel )
3		simp3 1002	. . 3 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. (SubGrp ` R ) )
4		rnglidlabl.i	. . . 4 |- I = ( R ↾s U )
5	4	subgabl 13668	. . 3 |- ( ( R e. Abel /\ U e. (SubGrp ` R ) ) -> I e. Abel )
6	2, 3, 5	syl2anc 411	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Abel )
7		eqid 2205	. . . 4 |- ( 0g ` R ) = ( 0g ` R )
8	7	subg0cl 13518	. . 3 |- ( U e. (SubGrp ` R ) -> ( 0g ` R ) e. U )
9		rnglidlabl.l	. . . 4 |- L = (LIdeal ` R )
10	9, 4, 7	rnglidlmsgrp 14259	. . 3 |- ( ( R e. Rng /\ U e. L /\ ( 0g ` R ) e. U ) -> (mulGrp ` I ) e. Smgrp )
11	8, 10	syl3an3 1285	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> (mulGrp ` I ) e. Smgrp )
12		simpl1 1003	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> R e. Rng )
13	9, 4	lidlssbas 14239	. . . . . . . . 9 |- ( U e. L -> ( Base ` I ) C_ ( Base ` R ) )
14	13	sseld 3192	. . . . . . . 8 |- ( U e. L -> ( a e. ( Base ` I ) -> a e. ( Base ` R ) ) )
15	13	sseld 3192	. . . . . . . 8 |- ( U e. L -> ( b e. ( Base ` I ) -> b e. ( Base ` R ) ) )
16	13	sseld 3192	. . . . . . . 8 |- ( U e. L -> ( c e. ( Base ` I ) -> c e. ( Base ` R ) ) )
17	14, 15, 16	3anim123d 1332	. . . . . . 7 |- ( U e. L -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
18	17	3ad2ant2 1022	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) )
19	18	imp 124	. . . . 5 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) )
20		eqid 2205	. . . . . 6 |- ( Base ` R ) = ( Base ` R )
21		eqid 2205	. . . . . 6 |- ( +g ` R ) = ( +g ` R )
22		eqid 2205	. . . . . 6 |- ( .r ` R ) = ( .r ` R )
23	20, 21, 22	rngdi 13702	. . . . 5 |- ( ( R e. Rng /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) -> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
24	12, 19, 23	syl2anc 411	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
25	20, 21, 22	rngdir 13703	. . . . 5 |- ( ( R e. Rng /\ ( a e. ( Base ` R ) /\ b e. ( Base ` R ) /\ c e. ( Base ` R ) ) ) -> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
26	12, 19, 25	syl2anc 411	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
27		simp2 1001	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> U e. L )
28		simp1 1000	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> R e. Rng )
29	4, 22	ressmulrg 12977	. . . . . . . . . 10 |- ( ( U e. L /\ R e. Rng ) -> ( .r ` R ) = ( .r ` I ) )
30	27, 28, 29	syl2anc 411	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` R ) = ( .r ` I ) )
31	30	eqcomd 2211	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( .r ` I ) = ( .r ` R ) )
32		eqidd 2206	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> a = a )
33	4	a1i 9	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I = ( R ↾s U ) )
34		eqidd 2206	. . . . . . . . . . 11 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` R ) )
35	33, 34, 27, 28	ressplusgd 12961	. . . . . . . . . 10 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` R ) = ( +g ` I ) )
36	35	eqcomd 2211	. . . . . . . . 9 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( +g ` I ) = ( +g ` R ) )
37	36	oveqd 5961	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( +g ` I ) c ) = ( b ( +g ` R ) c ) )
38	31, 32, 37	oveq123d 5965	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( a ( .r ` R ) ( b ( +g ` R ) c ) ) )
39	31	oveqd 5961	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) b ) = ( a ( .r ` R ) b ) )
40	31	oveqd 5961	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( .r ` I ) c ) = ( a ( .r ` R ) c ) )
41	36, 39, 40	oveq123d 5965	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) )
42	38, 41	eqeq12d 2220	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) <-> ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) ) )
43	36	oveqd 5961	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( a ( +g ` I ) b ) = ( a ( +g ` R ) b ) )
44		eqidd 2206	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> c = c )
45	31, 43, 44	oveq123d 5965	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( +g ` R ) b ) ( .r ` R ) c ) )
46	31	oveqd 5961	. . . . . . . 8 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( b ( .r ` I ) c ) = ( b ( .r ` R ) c ) )
47	36, 40, 46	oveq123d 5965	. . . . . . 7 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) )
48	45, 47	eqeq12d 2220	. . . . . 6 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) <-> ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) )
49	42, 48	anbi12d 473	. . . . 5 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> ( ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
50	49	adantr 276	. . . 4 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) <-> ( ( a ( .r ` R ) ( b ( +g ` R ) c ) ) = ( ( a ( .r ` R ) b ) ( +g ` R ) ( a ( .r ` R ) c ) ) /\ ( ( a ( +g ` R ) b ) ( .r ` R ) c ) = ( ( a ( .r ` R ) c ) ( +g ` R ) ( b ( .r ` R ) c ) ) ) ) )
51	24, 26, 50	mpbir2and 947	. . 3 |- ( ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) /\ ( a e. ( Base ` I ) /\ b e. ( Base ` I ) /\ c e. ( Base ` I ) ) ) -> ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) )
52	51	ralrimivvva 2589	. 2 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) )
53		eqid 2205	. . 3 |- ( Base ` I ) = ( Base ` I )
54		eqid 2205	. . 3 |- (mulGrp ` I ) = (mulGrp ` I )
55		eqid 2205	. . 3 |- ( +g ` I ) = ( +g ` I )
56		eqid 2205	. . 3 |- ( .r ` I ) = ( .r ` I )
57	53, 54, 55, 56	isrng 13696	. 2 |- ( I e. Rng <-> ( I e. Abel /\ (mulGrp ` I ) e. Smgrp /\ A. a e. ( Base ` I ) A. b e. ( Base ` I ) A. c e. ( Base ` I ) ( ( a ( .r ` I ) ( b ( +g ` I ) c ) ) = ( ( a ( .r ` I ) b ) ( +g ` I ) ( a ( .r ` I ) c ) ) /\ ( ( a ( +g ` I ) b ) ( .r ` I ) c ) = ( ( a ( .r ` I ) c ) ( +g ` I ) ( b ( .r ` I ) c ) ) ) ) )
58	6, 11, 52, 57	syl3anbrc 1184	1 |- ( ( R e. Rng /\ U e. L /\ U e. (SubGrp ` R ) ) -> I e. Rng )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2176   A.wral 2484   ` cfv 5271  (class class class)co 5944   Basecbs 12832   ↾_s cress 12833   +g cplusg 12909   .rcmulr 12910   0gc0g 13088  Smgrpcsgrp 13233  SubGrpcsubg 13503   Abelcabl 13621  mulGrpcmgp 13682  Rngcrng 13694  LIdealclidl 14229

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 615  ax-in2 616  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-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-addass 8027  ax-i2m1 8030  ax-0lt1 8031  ax-0id 8033  ax-rnegex 8034  ax-pre-ltirr 8037  ax-pre-lttrn 8039  ax-pre-ltadd 8041

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  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-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  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-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-pnf 8109  df-mnf 8110  df-ltxr 8112  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-iress 12840  df-plusg 12922  df-mulr 12923  df-sca 12925  df-vsca 12926  df-ip 12927  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-subg 13506  df-cmn 13622  df-abl 13623  df-mgp 13683  df-rng 13695  df-lssm 14115  df-sra 14197  df-rgmod 14198  df-lidl 14231

This theorem is referenced by:  rng2idlsubgsubrng  14282