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Theorem dvdsrmul 15446
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 443 . 2  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  e.  B )
2 simpr 447 . . 3  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  Y  e.  B )
3 eqid 2296 . . 3  |-  ( Y 
.x.  X )  =  ( Y  .x.  X
)
4 oveq1 5881 . . . . 5  |-  ( z  =  Y  ->  (
z  .x.  X )  =  ( Y  .x.  X ) )
54eqeq1d 2304 . . . 4  |-  ( z  =  Y  ->  (
( z  .x.  X
)  =  ( Y 
.x.  X )  <->  ( Y  .x.  X )  =  ( Y  .x.  X ) ) )
65rspcev 2897 . . 3  |-  ( ( Y  e.  B  /\  ( Y  .x.  X )  =  ( Y  .x.  X ) )  ->  E. z  e.  B  ( z  .x.  X
)  =  ( Y 
.x.  X ) )
72, 3, 6sylancl 643 . 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 15444 . 2  |-  ( X 
.||  ( Y  .x.  X )  <->  ( X  e.  B  /\  E. z  e.  B  ( z  .x.  X )  =  ( Y  .x.  X ) ) )
121, 7, 11sylanbrc 645 1  |-  ( ( X  e.  B  /\  Y  e.  B )  ->  X  .||  ( Y  .x.  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   ||rcdsr 15436
This theorem is referenced by:  dvdsrid  15449  dvdsrtr  15450  dvdsrmul1  15451  dvdsrneg  15452  unitmulclb  15463  unitgrp  15465  isdrng2  15538  subrguss  15576  subrgunit  15579  fidomndrnglem  16063  dvdsq1p  19562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-dvdsr 15439
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