Theorem dvdsr02 14118

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 14059	. . . 4 |- ( R e. Ring -> R e. SRing )
6	5	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
7		eqid 2231	. . . 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 14033	. . . 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 14108	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> E. x e. B ( x ( .r ` R ) .0. ) = X ) )
13	1, 7, 9	ringrz 14056	. . . . . . 7 |- ( ( R e. Ring /\ x e. B ) -> ( x ( .r ` R ) .0. ) = .0. )
14	13	eqeq1d 2240	. . . . . 6 |- ( ( R e. Ring /\ x e. B ) -> ( ( x ( .r ` R ) .0. ) = X <-> .0. = X ) )
15		eqcom 2233	. . . . . 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 2529	. . . 4 |- ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> E. x e. B X = .0. ) )
18		elex2 2819	. . . . 5 |- ( .0. e. B -> E. w w e. B )
19		r19.9rmv 3586	. . . . 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 1397   E.wex 1540   e. wcel 2202   E.wrex 2511   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   Basecbs 13081   .rcmulr 13160   0gc0g 13338  SRingcsrg 13975   Ringcrg 14008   ||_rcdsr 14098

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-pre-ltirr 8143  ax-pre-ltadd 8147

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 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-inn 9143  df-2 9201  df-3 9202  df-ndx 13084  df-slot 13085  df-base 13087  df-sets 13088  df-plusg 13172  df-mulr 13173  df-0g 13340  df-mgm 13438  df-sgrp 13484  df-mnd 13499  df-grp 13585  df-minusg 13586  df-cmn 13872  df-abl 13873  df-mgp 13933  df-ur 13972  df-srg 13976  df-ring 14010  df-dvdsr 14101

This theorem is referenced by: (None)