Theorem dvdsr01 14121

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 14037	. . 3 |- ( R e. Ring -> .0. e. B )
4		eqid 2231	. . . 4 |- ( .r ` R ) = ( .r ` R )
5	1, 4, 2	ringlz 14059	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( .r ` R ) X ) = .0. )
6		oveq1 6025	. . . . 5 |- ( x = .0. -> ( x ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
7	6	eqeq1d 2240	. . . 4 |- ( x = .0. -> ( ( x ( .r ` R ) X ) = .0. <-> ( .0. ( .r ` R ) X ) = .0. ) )
8	7	rspcev 2910	. . 3 |- ( ( .0. e. B /\ ( .0. ( .r ` R ) X ) = .0. ) -> E. x e. B ( x ( .r ` R ) X ) = .0. )
9	3, 5, 8	syl2an2r 599	. 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 14063	. . . 4 |- ( R e. Ring -> R e. SRing )
14	13	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
15		eqidd 2232	. . 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 14112	. 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 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   Basecbs 13084   .rcmulr 13163   0gc0g 13341  SRingcsrg 13979   Ringcrg 14012   ||_rcdsr 14102

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-cmn 13875  df-abl 13876  df-mgp 13937  df-ur 13976  df-srg 13980  df-ring 14014  df-dvdsr 14105

This theorem is referenced by: (None)