Theorem rnglz 13707

Description: The zero of a non-unital ring is a left-absorbing element. (Contributed by FL, 31-Aug-2009.) Generalization of ringlz 13805. (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 13697	. . . . . . 7 |- ( R e. Rng -> R e. Abel )
2		ablgrp 13625	. . . . . . 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 13361	. . . . . 6 |- ( R e. Grp -> .0. e. B )
7		eqid 2205	. . . . . . 7 |- ( +g ` R ) = ( +g ` R )
8	4, 7, 5	grplid 13363	. . . . . 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 5959	. . 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 983	. . . . 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 13703	. . . 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 13706	. . . . 5 |- ( ( R e. Rng /\ .0. e. B /\ X e. B ) -> ( .0. .x. X ) e. B )
25	12, 22, 23, 24	syl3anc 1250	. . . 4 |- ( ( R e. Rng /\ X e. B ) -> ( .0. .x. X ) e. B )
26	4, 7, 5	grprid 13364	. . . . 5 |- ( ( R e. Grp /\ ( .0. .x. X ) e. B ) -> ( ( .0. .x. X ) ( +g ` R ) .0. ) = ( .0. .x. X ) )
27	26	eqcomd 2211	. . . 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 2246	. 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 13394	. . 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 1252	. 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 981   = wceq 1373   e. wcel 2176   ` cfv 5271  (class class class)co 5944   Basecbs 12832   +g cplusg 12909   .rcmulr 12910   0gc0g 13088   Grpcgrp 13332   Abelcabl 13621  Rngcrng 13694

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-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-ndx 12835  df-slot 12836  df-base 12838  df-sets 12839  df-plusg 12922  df-mulr 12923  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-abl 13623  df-mgp 13683  df-rng 13695

This theorem is referenced by:  rngmneg1  13709