Theorem dvdsr01 14520

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 14433	. . 3 |- ( R e. Ring -> .0. e. B )
4		eqid 2238	. . . 4 |- ( .r ` R ) = ( .r ` R )
5	1, 4, 2	ringlz 14455	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .0. ( .r ` R ) X ) = .0. )
6		oveq1 6095	. . . . 5 |- ( x = .0. -> ( x ( .r ` R ) X ) = ( .0. ( .r ` R ) X ) )
7	6	eqeq1d 2247	. . . 4 |- ( x = .0. -> ( ( x ( .r ` R ) X ) = .0. <-> ( .0. ( .r ` R ) X ) = .0. ) )
8	7	rspcev 2929	. . 3 |- ( ( .0. e. B /\ ( .0. ( .r ` R ) X ) = .0. ) -> E. x e. B ( x ( .r ` R ) X ) = .0. )
9	3, 5, 8	syl2an2r 603	. 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 14459	. . . 4 |- ( R e. Ring -> R e. SRing )
14	13	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
15		eqidd 2239	. . 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 14511	. 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. )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   Basecbs 13410   .rcmulr 13491   0gc0g 13669  SRingcsrg 14374   Ringcrg 14407   ||_rcdsr 14501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-pnf 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-dvdsr 14504

This theorem is used by: (None)