Theorem dvdsr02 14350

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 14290	. . . 4 |- ( R e. Ring -> R e. SRing )
6	5	adantr 276	. . 3 |- ( ( R e. Ring /\ X e. B ) -> R e. SRing )
7		eqid 2234	. . . 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 14264	. . . 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 14340	. 2 |- ( ( R e. Ring /\ X e. B ) -> ( .0. .|| X <-> E. x e. B ( x ( .r ` R ) .0. ) = X ) )
13	1, 7, 9	ringrz 14287	. . . . . . 7 |- ( ( R e. Ring /\ x e. B ) -> ( x ( .r ` R ) .0. ) = .0. )
14	13	eqeq1d 2243	. . . . . 6 |- ( ( R e. Ring /\ x e. B ) -> ( ( x ( .r ` R ) .0. ) = X <-> .0. = X ) )
15		eqcom 2236	. . . . . 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 2541	. . . 4 |- ( R e. Ring -> ( E. x e. B ( x ( .r ` R ) .0. ) = X <-> E. x e. B X = .0. ) )
18		elex2 2832	. . . . 5 |- ( .0. e. B -> E. w w e. B )
19		r19.9rmv 3605	. . . . 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 2205   E.wrex 2523   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   Basecbs 13296   .rcmulr 13375   0gc0g 13553  SRingcsrg 14206   Ringcrg 14239   ||_rcdsr 14330

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-ltadd 8259

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-pnf 8326  df-mnf 8327  df-ltxr 8329  df-inn 9255  df-2 9313  df-3 9314  df-ndx 13299  df-slot 13300  df-base 13302  df-sets 13303  df-plusg 13387  df-mulr 13388  df-0g 13555  df-mgm 13619  df-sgrp 13665  df-mnd 13678  df-grp 13758  df-minusg 13759  df-cmn 14039  df-abl 14040  df-mgp 14160  df-ur 14203  df-srg 14207  df-ring 14241  df-dvdsr 14333

This theorem is referenced by: (None)