Theorem dvdsr02 14200

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 14141	. . . 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 14115	. . . 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 14190	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> E. x e. B ( x ( .r ` R ) .0. ) = X ) )
13	1, 7, 9	ringrz 14138	. . . . . . 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 2530	. . . 4 |- ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> E. x e. B X = .0. ) )
18		elex2 2820	. . . . 5 |- ( .0. e. B -> E. w w e. B )
19		r19.9rmv 3588	. . . . 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 1398   E.wex 1541   e. wcel 2202   E.wrex 2512   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   Basecbs 13162   .rcmulr 13241   0gc0g 13419  SRingcsrg 14057   Ringcrg 14090   ||_rcdsr 14180

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-addcom 8192  ax-addass 8194  ax-i2m1 8197  ax-0lt1 8198  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-ltadd 8208

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-ltxr 8278  df-inn 9203  df-2 9261  df-3 9262  df-ndx 13165  df-slot 13166  df-base 13168  df-sets 13169  df-plusg 13253  df-mulr 13254  df-0g 13421  df-mgm 13519  df-sgrp 13565  df-mnd 13580  df-grp 13666  df-minusg 13667  df-cmn 13953  df-abl 13954  df-mgp 14015  df-ur 14054  df-srg 14058  df-ring 14092  df-dvdsr 14183

This theorem is referenced by: (None)