Theorem rnglz 14039

Description: The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 14137. (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 14029	. . . . . . 7 |- ( R e. Rng -> R e. Abel )
2		ablgrp 13956	. . . . . . 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 13692	. . . . . 6 |- ( R e. Grp -> .0. e. B )
7		eqid 2231	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
8	4, 7, 5	grplid 13694	. . . . . 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 6043	. . 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 14035	. . . 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 14038	. . . . 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 13695	. . . . 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 13725	. . 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 2202   ` cfv 5333  (class class class)co 6028   Basecbs 13162   +g cplusg 13240   .rcmulr 13241   0gc0g 13419   Grpcgrp 13663   Abelcabl 13952  Rngcrng 14026

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-abl 13954  df-mgp 14015  df-rng 14027

This theorem is referenced by:  rngmneg1  14041