Theorem dvdsr01 13204

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 13135	. . 3 |- ( R e. Ring -> .0. e. B )
4		eqid 2177	. . . 4 |- ( .r ` R ) = ( .r ` R )
5	1, 4, 2	ringlz 13153	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( .r ` R ) X ) = .0. )
6		oveq1 5879	. . . . 5 |- ( x = .0. -> ( x ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
7	6	eqeq1d 2186	. . . 4 |- ( x = .0. -> ( ( x ( .r ` R ) X ) = .0. <-> ( .0. ( .r ` R ) X ) = .0. ) )
8	7	rspcev 2841	. . 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 13155	. . . 4 |- ( R e. Ring -> R e. SRing )
14	13	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
15		eqidd 2178	. . 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 13195	. 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 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4002   ` cfv 5215  (class class class)co 5872   Basecbs 12454   .rcmulr 12529   0gc0g 12693  SRingcsrg 13077   Ringcrg 13110   ||_rcdsr 13186

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7899  ax-resscn 7900  ax-1cn 7901  ax-1re 7902  ax-icn 7903  ax-addcl 7904  ax-addrcl 7905  ax-mulcl 7906  ax-addcom 7908  ax-addass 7910  ax-i2m1 7913  ax-0lt1 7914  ax-0id 7916  ax-rnegex 7917  ax-pre-ltirr 7920  ax-pre-ltadd 7924

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-f 5219  df-f1 5220  df-fo 5221  df-f1o 5222  df-fv 5223  df-riota 5828  df-ov 5875  df-oprab 5876  df-mpo 5877  df-pnf 7990  df-mnf 7991  df-ltxr 7993  df-inn 8916  df-2 8974  df-3 8975  df-ndx 12457  df-slot 12458  df-base 12460  df-sets 12461  df-plusg 12541  df-mulr 12542  df-0g 12695  df-mgm 12707  df-sgrp 12740  df-mnd 12750  df-grp 12812  df-minusg 12813  df-cmn 13021  df-abl 13022  df-mgp 13062  df-ur 13074  df-srg 13078  df-ring 13112  df-dvdsr 13189

This theorem is referenced by: (None)