Theorem rnglz 13961

Description: The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 14059. (Revised by AV, 17-Apr-2020.)

Hypotheses

Ref	Expression
rngcl.b	|- B = ( Base ` R )
rngcl.t	|- .x. = ( .r ` R )
rnglz.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
rnglz	|- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. )

Proof of Theorem rnglz

Step	Hyp	Ref	Expression
1		rngabl 13951	. . . . . . 7 |- ( R e. Rng -> R e. Abel )
2		ablgrp 13878	. . . . . . 7 |- ( R e. Abel -> R e. Grp )
3	1, 2	syl 14	. . . . . 6 |- ( R e. Rng -> R e. Grp )
4		rngcl.b	. . . . . . 7 |- B = ( Base ` R )
5		rnglz.z	. . . . . . 7 |- .0. = ( 0g ` R )
6	4, 5	grpidcl 13614	. . . . . 6 |- ( R e. Grp -> .0. e. B )
7		eqid 2231	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
8	4, 7, 5	grplid 13616	. . . . . 6 |- ( ( R e. Grp /\ .0. e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
9	3, 6, 8	syl2anc2 412	. . . . 5 |- ( R e. Rng -> ( .0. ( +g ` R ) .0. ) = .0. )
10	9	adantr 276	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. ( +g ` R ) .0. ) = .0. )
11	10	oveq1d 6033	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( .0. .x. X ) )
12		simpl 109	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> R e. Rng )
13	3, 6	syl 14	. . . . . . 7 |- ( R e. Rng -> .0. e. B )
14	13, 13	jca 306	. . . . . 6 |- ( R e. Rng -> ( .0. e. B /\ .0. e. B ) )
15	14	anim1i 340	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. e. B /\ .0. e. B ) /\ X e. B ) )
16		df-3an 1006	. . . . 5 |- ( ( .0. e. B /\ .0. e. B /\ X e. B ) <-> ( ( .0. e. B /\ .0. e. B ) /\ X e. B ) )
17	15, 16	sylibr 134	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. e. B /\ .0. e. B /\ X e. B ) )
18		rngcl.t	. . . . 5 |- .x. = ( .r ` R )
19	4, 7, 18	rngdir 13957	. . . 4 |- ( ( R e. Rng /\ ( .0. e. B /\ .0. e. B /\ X e. B ) ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) )
20	12, 17, 19	syl2anc 411	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. ( +g ` R ) .0. ) .x. X ) = ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) )
21	3	adantr 276	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> R e. Grp )
22	13	adantr 276	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> .0. e. B )
23		simpr 110	. . . . 5 |- ( ( R e. Rng /\ X e. B ) -> X e. B )
24	4, 18	rngcl 13960	. . . . 5 |- ( ( R e. Rng /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
25	12, 22, 23, 24	syl3anc 1273	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) e. B )
26	4, 7, 5	grprid 13617	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
27	26	eqcomd 2237	. . . 4 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( .0. .x. X ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
28	21, 25, 27	syl2anc 411	. . 3 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
29	11, 20, 28	3eqtr3d 2272	. 2 |- ( ( R e. Rng /\ X e. B ) -> ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) )
30	4, 7	grplcan 13647	. . 3 |- ( ( R e. Grp /\ ( ( .0. .x. X ) e. B /\ .0. e. B /\ ( .0. .x. X ) e. B ) ) -> ( ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) <-> ( .0. .x. X ) = .0. ) )
31	21, 25, 22, 25, 30	syl13anc 1275	. 2 |- ( ( R e. Rng /\ X e. B ) -> ( ( ( .0. .x. X ) ( +g ` R ) ( .0. .x. X ) ) = ( ( .0. .x. X ) ( +g ` R ) .0. ) <-> ( .0. .x. X ) = .0. ) )
32	29, 31	mpbid 147	1 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) = .0. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6018   Basecbs 13084   +g cplusg 13162   .rcmulr 13163   0gc0g 13341   Grpcgrp 13585   Abelcabl 13874  Rngcrng 13948

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-addass 8134  ax-i2m1 8137  ax-0lt1 8138  ax-0id 8140  ax-rnegex 8141  ax-pre-ltirr 8144  ax-pre-ltadd 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-pnf 8216  df-mnf 8217  df-ltxr 8219  df-inn 9144  df-2 9202  df-3 9203  df-ndx 13087  df-slot 13088  df-base 13090  df-sets 13091  df-plusg 13175  df-mulr 13176  df-0g 13343  df-mgm 13441  df-sgrp 13487  df-mnd 13502  df-grp 13588  df-minusg 13589  df-abl 13876  df-mgp 13937  df-rng 13949

This theorem is referenced by:  rngmneg1  13963