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Theorem dvdsrpropd 17124
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 648 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
32eqeq1d 2464 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
( x ( .r
`  K ) y )  =  z  <->  ( x
( .r `  L
) y )  =  z ) )
43an32s 802 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( .r
`  K ) y )  =  z  <->  ( x
( .r `  L
) y )  =  z ) )
54rexbidva 2965 . . . . 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 641 . . . 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 2532 . . . . 5  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  K
) ) )
97rexeqdv 3060 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( .r `  K ) y )  =  z  <->  E. x  e.  ( Base `  K ) ( x ( .r `  K ) y )  =  z ) )
108, 9anbi12d 710 . . . 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 2532 . . . . 5  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  L
) ) )
1311rexeqdv 3060 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( .r `  L ) y )  =  z  <->  E. x  e.  ( Base `  L ) ( x ( .r `  L ) y )  =  z ) )
1412, 13anbi12d 710 . . . 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 283 . . 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 4505 . 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 2462 . . 3  |-  ( Base `  K )  =  (
Base `  K )
18 eqid 2462 . . 3  |-  ( ||r `  K
)  =  ( ||r `  K
)
19 eqid 2462 . . 3  |-  ( .r
`  K )  =  ( .r `  K
)
2017, 18, 19dvdsrval 17073 . 2  |-  ( ||r `  K
)  =  { <. y ,  z >.  |  ( y  e.  ( Base `  K )  /\  E. x  e.  ( Base `  K ) ( x ( .r `  K
) y )  =  z ) }
21 eqid 2462 . . 3  |-  ( Base `  L )  =  (
Base `  L )
22 eqid 2462 . . 3  |-  ( ||r `  L
)  =  ( ||r `  L
)
23 eqid 2462 . . 3  |-  ( .r
`  L )  =  ( .r `  L
)
2421, 22, 23dvdsrval 17073 . 2  |-  ( ||r `  L
)  =  { <. y ,  z >.  |  ( y  e.  ( Base `  L )  /\  E. x  e.  ( Base `  L ) ( x ( .r `  L
) y )  =  z ) }
2516, 20, 243eqtr4g 2528 1  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   E.wrex 2810   {copab 4499   ` cfv 5581  (class class class)co 6277   Basecbs 14481   .rcmulr 14547   ||rcdsr 17066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-dvdsr 17069
This theorem is referenced by:  unitpropd  17125
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