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Theorem dvdsrpropd 15478
Description: The divisibility relation depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.)
Hypotheses
Ref Expression
rngidpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
rngidpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
rngidpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
Assertion
Ref Expression
dvdsrpropd  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
Distinct variable groups:    x, y, B    x, K, y    x, L, y    ph, x, y

Proof of Theorem dvdsrpropd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 rngidpropd.3 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
21anassrs 629 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
32eqeq1d 2291 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
( x ( .r
`  K ) y )  =  z  <->  ( x
( .r `  L
) y )  =  z ) )
43an32s 779 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( .r
`  K ) y )  =  z  <->  ( x
( .r `  L
) y )  =  z ) )
54rexbidva 2560 . . . . 5  |-  ( (
ph  /\  y  e.  B )  ->  ( E. x  e.  B  ( x ( .r
`  K ) y )  =  z  <->  E. x  e.  B  ( x
( .r `  L
) y )  =  z ) )
65pm5.32da 622 . . . 4  |-  ( ph  ->  ( ( y  e.  B  /\  E. x  e.  B  ( x
( .r `  K
) y )  =  z )  <->  ( y  e.  B  /\  E. x  e.  B  ( x
( .r `  L
) y )  =  z ) ) )
7 rngidpropd.1 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  K ) )
87eleq2d 2350 . . . . 5  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  K
) ) )
97rexeqdv 2743 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( .r `  K ) y )  =  z  <->  E. x  e.  ( Base `  K ) ( x ( .r `  K ) y )  =  z ) )
108, 9anbi12d 691 . . . 4  |-  ( ph  ->  ( ( y  e.  B  /\  E. x  e.  B  ( x
( .r `  K
) y )  =  z )  <->  ( y  e.  ( Base `  K
)  /\  E. x  e.  ( Base `  K
) ( x ( .r `  K ) y )  =  z ) ) )
11 rngidpropd.2 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  L ) )
1211eleq2d 2350 . . . . 5  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  L
) ) )
1311rexeqdv 2743 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( .r `  L ) y )  =  z  <->  E. x  e.  ( Base `  L ) ( x ( .r `  L ) y )  =  z ) )
1412, 13anbi12d 691 . . . 4  |-  ( ph  ->  ( ( y  e.  B  /\  E. x  e.  B  ( x
( .r `  L
) y )  =  z )  <->  ( y  e.  ( Base `  L
)  /\  E. x  e.  ( Base `  L
) ( x ( .r `  L ) y )  =  z ) ) )
156, 10, 143bitr3d 274 . . 3  |-  ( ph  ->  ( ( y  e.  ( Base `  K
)  /\  E. x  e.  ( Base `  K
) ( x ( .r `  K ) y )  =  z )  <->  ( y  e.  ( Base `  L
)  /\  E. x  e.  ( Base `  L
) ( x ( .r `  L ) y )  =  z ) ) )
1615opabbidv 4082 . 2  |-  ( ph  ->  { <. y ,  z
>.  |  ( y  e.  ( Base `  K
)  /\  E. x  e.  ( Base `  K
) ( x ( .r `  K ) y )  =  z ) }  =  { <. y ,  z >.  |  ( y  e.  ( Base `  L
)  /\  E. x  e.  ( Base `  L
) ( x ( .r `  L ) y )  =  z ) } )
17 eqid 2283 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2283 . . 3  |-  ( ||r `  K
)  =  ( ||r `  K
)
19 eqid 2283 . . 3  |-  ( .r
`  K )  =  ( .r `  K
)
2017, 18, 19dvdsrval 15427 . 2  |-  ( ||r `  K
)  =  { <. y ,  z >.  |  ( y  e.  ( Base `  K )  /\  E. x  e.  ( Base `  K ) ( x ( .r `  K
) y )  =  z ) }
21 eqid 2283 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2283 . . 3  |-  ( ||r `  L
)  =  ( ||r `  L
)
23 eqid 2283 . . 3  |-  ( .r
`  L )  =  ( .r `  L
)
2421, 22, 23dvdsrval 15427 . 2  |-  ( ||r `  L
)  =  { <. y ,  z >.  |  ( y  e.  ( Base `  L )  /\  E. x  e.  ( Base `  L ) ( x ( .r `  L
) y )  =  z ) }
2516, 20, 243eqtr4g 2340 1  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   {copab 4076   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   ||rcdsr 15420
This theorem is referenced by:  unitpropd  15479
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-dvdsr 15423
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