Theorem dvdsr02 14069

Description: Only zero is divisible by zero. (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
dvdsr02	|- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> X = .0. ) )

Proof of Theorem dvdsr02

Dummy variables x w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdsr0.b	. . . 4 |- B = ( Base ` R )
2	1	a1i 9	. . 3 |- ( ( R e. Ring /\ X e. B ) -> B = ( Base ` R ) )
3		dvdsr0.d	. . . 4 |- .|| = ( ||r ` R )
4	3	a1i 9	. . 3 |- ( ( R e. Ring /\ X e. B ) -> .|| = ( ||r ` R ) )
5		ringsrg 14010	. . . 4 |- ( R e. Ring -> R e. SRing )
6	5	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
7		eqid 2229	. . . 4 |- ( .r ` R ) = ( .r ` R )
8	7	a1i 9	. . 3 |- ( ( R e. Ring /\ X e. B ) -> ( .r ` R ) = ( .r ` R ) )
9		dvdsr0.z	. . . . 5 |- .0. = ( 0g ` R )
10	1, 9	ring0cl 13984	. . . 4 |- ( R e. Ring -> .0. e. B )
11	10	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> .0. e. B )
12	2, 4, 6, 8, 11	dvdsr2d 14059	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> E. x e. B ( x ( .r ` R ) .0. ) = X ) )
13	1, 7, 9	ringrz 14007	. . . . . . 7 |- ( ( R e. Ring /\ x e. B ) -> ( x ( .r ` R ) .0. ) = .0. )
14	13	eqeq1d 2238	. . . . . 6 |- ( ( R e. Ring /\ x e. B ) -> ( ( x ( .r ` R ) .0. ) = X <-> .0. = X ) )
15		eqcom 2231	. . . . . 6 |- ( .0. = X <-> X = .0. )
16	14, 15	bitrdi 196	. . . . 5 |- ( ( R e. Ring /\ x e. B ) -> ( ( x ( .r ` R ) .0. ) = X <-> X = .0. ) )
17	16	rexbidva 2527	. . . 4 |- ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> E. x e. B X = .0. ) )
18		elex2 2816	. . . . 5 |- ( .0. e. B -> E. w w e. B )
19		r19.9rmv 3583	. . . . 5 |- ( E. w w e. B -> ( X = .0. <-> E. x e. B X = .0. ) )
20	10, 18, 19	3syl 17	. . . 4 |- ( R e. Ring -> ( X = .0. <-> E. x e. B X = .0. ) )
21	17, 20	bitr4d 191	. . 3 |- ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> X = .0. ) )
22	21	adantr 276	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> X = .0. ) )
23	12, 22	bitrd 188	1 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> X = .0. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   class class class wbr 4083   ` cfv 5318  (class class class)co 6001   Basecbs 13032   .rcmulr 13111   0gc0g 13289  SRingcsrg 13926   Ringcrg 13959   ||_rcdsr 14049

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-pre-ltirr 8111  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-ltxr 8186  df-inn 9111  df-2 9169  df-3 9170  df-ndx 13035  df-slot 13036  df-base 13038  df-sets 13039  df-plusg 13123  df-mulr 13124  df-0g 13291  df-mgm 13389  df-sgrp 13435  df-mnd 13450  df-grp 13536  df-minusg 13537  df-cmn 13823  df-abl 13824  df-mgp 13884  df-ur 13923  df-srg 13927  df-ring 13961  df-dvdsr 14052

This theorem is referenced by: (None)