Theorem dvdsr01 13738

Description: In a ring, zero is divisible by all elements. ("Zero divisor" as a term has a somewhat different meaning.) (Contributed by Stefan O'Rear, 29-Mar-2015.)

Hypotheses

Ref	Expression
dvdsr0.b	|- B = ( Base ` R )
dvdsr0.d	|- .|| = ( ||r ` R )
dvdsr0.z	|- .0. = ( 0g ` R )

Assertion

Ref	Expression
dvdsr01	|- ( ( R e. Ring /\ X e. B ) -> X .|| .0. )

Proof of Theorem dvdsr01

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr0.b	. . . 4 |- B = ( Base ` R )
2		dvdsr0.z	. . . 4 |- .0. = ( 0g ` R )
3	1, 2	ring0cl 13655	. . 3 |- ( R e. Ring -> .0. e. B )
4		eqid 2196	. . . 4 |- ( .r ` R ) = ( .r ` R )
5	1, 4, 2	ringlz 13677	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( .r ` R ) X ) = .0. )
6		oveq1 5932	. . . . 5 |- ( x = .0. -> ( x ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
7	6	eqeq1d 2205	. . . 4 |- ( x = .0. -> ( ( x ( .r ` R ) X ) = .0. <-> ( .0. ( .r ` R ) X ) = .0. ) )
8	7	rspcev 2868	. . 3 |- ( ( .0. e. B /\ ( .0. ( .r ` R ) X ) = .0. ) -> E. x e. B ( x ( .r ` R ) X ) = .0. )
9	3, 5, 8	syl2an2r 595	. 2 |- ( ( R e. Ring /\ X e. B ) -> E. x e. B ( x ( .r ` R ) X ) = .0. )
10	1	a1i 9	. . 3 |- ( ( R e. Ring /\ X e. B ) -> B = ( Base ` R ) )
11		dvdsr0.d	. . . 4 |- .|| = ( ||r ` R )
12	11	a1i 9	. . 3 |- ( ( R e. Ring /\ X e. B ) -> .|| = ( ||r ` R ) )
13		ringsrg 13681	. . . 4 |- ( R e. Ring -> R e. SRing )
14	13	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
15		eqidd 2197	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .r ` R ) = ( .r ` R ) )
16		simpr 110	. . 3 |- ( ( R e. Ring /\ X e. B ) -> X e. B )
17	10, 12, 14, 15, 16	dvdsr2d 13729	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( X .|| .0. <-> E. x e. B ( x ( .r ` R ) X ) = .0. ) )
18	9, 17	mpbird 167	1 |- ( ( R e. Ring /\ X e. B ) -> X .|| .0. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4034   ` cfv 5259  (class class class)co 5925   Basecbs 12705   .rcmulr 12783   0gc0g 12960  SRingcsrg 13597   Ringcrg 13630   ||_rcdsr 13720

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7989  ax-resscn 7990  ax-1cn 7991  ax-1re 7992  ax-icn 7993  ax-addcl 7994  ax-addrcl 7995  ax-mulcl 7996  ax-addcom 7998  ax-addass 8000  ax-i2m1 8003  ax-0lt1 8004  ax-0id 8006  ax-rnegex 8007  ax-pre-ltirr 8010  ax-pre-ltadd 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8082  df-mnf 8083  df-ltxr 8085  df-inn 9010  df-2 9068  df-3 9069  df-ndx 12708  df-slot 12709  df-base 12711  df-sets 12712  df-plusg 12795  df-mulr 12796  df-0g 12962  df-mgm 13060  df-sgrp 13106  df-mnd 13121  df-grp 13207  df-minusg 13208  df-cmn 13494  df-abl 13495  df-mgp 13555  df-ur 13594  df-srg 13598  df-ring 13632  df-dvdsr 13723

This theorem is referenced by: (None)