Theorem rnglz 14106

Description: The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 14204. (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 14096	. . . . . . 7 |- ( R e. Rng -> R e. Abel )
2		ablgrp 14023	. . . . . . 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 13759	. . . . . 6 |- ( R e. Grp -> .0. e. B )
7		eqid 2234	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
8	4, 7, 5	grplid 13761	. . . . . 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 6067	. . 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 1007	. . . . 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 14102	. . . 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 14105	. . . . 5 |- ( ( R e. Rng /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
25	12, 22, 23, 24	syl3anc 1274	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) e. B )
26	4, 7, 5	grprid 13762	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
27	26	eqcomd 2240	. . . 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 2275	. 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 13792	. . 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 1276	. 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 1005   = wceq 1398   e. wcel 2205   ` cfv 5354  (class class class)co 6052   Basecbs 13229   +g cplusg 13307   .rcmulr 13308   0gc0g 13486   Grpcgrp 13730   Abelcabl 14019  Rngcrng 14093

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-pre-ltirr 8241  ax-pre-ltadd 8245

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-ltxr 8315  df-inn 9240  df-2 9298  df-3 9299  df-ndx 13232  df-slot 13233  df-base 13235  df-sets 13236  df-plusg 13320  df-mulr 13321  df-0g 13488  df-mgm 13586  df-sgrp 13632  df-mnd 13647  df-grp 13733  df-minusg 13734  df-abl 14021  df-mgp 14082  df-rng 14094

This theorem is referenced by:  rngmneg1  14108