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Theorem dvdsrmul 15754
Description: A left-multiple of  X is divisible by  X. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
dvdsr.1  |-  B  =  ( Base `  R
)
dvdsr.2  |-  .||  =  (
||r `  R )
dvdsr.3  |-  .x.  =  ( .r `  R )
Assertion
Ref Expression
dvdsrmul  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  .||  ( Y  .x.  X ) )

Proof of Theorem dvdsrmul
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 simpl 445 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
2 simpr 449 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
3 eqid 2437 . . 3  |-  ( Y 
.x.  X )  =  ( Y  .x.  X
)
4 oveq1 6089 . . . . 5  |-  ( z  =  Y  ->  (
z  .x.  X )  =  ( Y  .x.  X ) )
54eqeq1d 2445 . . . 4  |-  ( z  =  Y  ->  (
( z  .x.  X
)  =  ( Y 
.x.  X )  <->  ( Y  .x.  X )  =  ( Y  .x.  X ) ) )
65rspcev 3053 . . 3  |-  ( ( Y  e.  B  /\  ( Y  .x.  X )  =  ( Y  .x.  X ) )  ->  E. z  e.  B  ( z  .x.  X
)  =  ( Y 
.x.  X ) )
72, 3, 6sylancl 645 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  E. z  e.  B  ( z  .x.  X
)  =  ( Y 
.x.  X ) )
8 dvdsr.1 . . 3  |-  B  =  ( Base `  R
)
9 dvdsr.2 . . 3  |-  .||  =  (
||r `  R )
10 dvdsr.3 . . 3  |-  .x.  =  ( .r `  R )
118, 9, 10dvdsr 15752 . 2  |-  ( X 
.||  ( Y  .x.  X )  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  ( Y  .x.  X ) ) )
121, 7, 11sylanbrc 647 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  .||  ( Y  .x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   E.wrex 2707   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   Basecbs 13470   .rcmulr 13531   ||rcdsr 15744
This theorem is referenced by:  dvdsrid  15757  dvdsrtr  15758  dvdsrmul1  15759  dvdsrneg  15760  unitmulclb  15771  unitgrp  15773  isdrng2  15846  subrguss  15884  subrgunit  15887  fidomndrnglem  16367  dvdsq1p  20084
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-dvdsr 15747
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