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Theorem dvdsrpropdg 13269
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
dvdsrpropdg.1  |-  ( ph  ->  B  =  ( Base `  K ) )
dvdsrpropdg.2  |-  ( ph  ->  B  =  ( Base `  L ) )
dvdsrpropdg.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
dvdsrpropdg.k  |-  ( ph  ->  K  e. SRing )
dvdsrpropdg.l  |-  ( ph  ->  L  e. SRing )
Assertion
Ref Expression
dvdsrpropdg  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
Distinct variable groups:    x, y, B   
x, K, y    x, L, y    ph, x, y

Proof of Theorem dvdsrpropdg
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 dvdsrpropdg.3 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
21anassrs 400 . . . . . . . 8  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
x ( .r `  K ) y )  =  ( x ( .r `  L ) y ) )
32eqeq1d 2186 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  B )  ->  (
( x ( .r
`  K ) y )  =  z  <->  ( x
( .r `  L
) y )  =  z ) )
43an32s 568 . . . . . 6  |-  ( ( ( ph  /\  y  e.  B )  /\  x  e.  B )  ->  (
( x ( .r
`  K ) y )  =  z  <->  ( x
( .r `  L
) y )  =  z ) )
54rexbidva 2474 . . . . 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 452 . . . 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 dvdsrpropdg.1 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  K ) )
87eleq2d 2247 . . . . 5  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  K
) ) )
97rexeqdv 2679 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( .r `  K ) y )  =  z  <->  E. x  e.  ( Base `  K ) ( x ( .r `  K ) y )  =  z ) )
108, 9anbi12d 473 . . . 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 dvdsrpropdg.2 . . . . . 6  |-  ( ph  ->  B  =  ( Base `  L ) )
1211eleq2d 2247 . . . . 5  |-  ( ph  ->  ( y  e.  B  <->  y  e.  ( Base `  L
) ) )
1311rexeqdv 2679 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ( x ( .r `  L ) y )  =  z  <->  E. x  e.  ( Base `  L ) ( x ( .r `  L ) y )  =  z ) )
1412, 13anbi12d 473 . . . 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 218 . . 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 4069 . 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 eqidd 2178 . . 3  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  K ) )
18 eqidd 2178 . . 3  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 K ) )
19 dvdsrpropdg.k . . 3  |-  ( ph  ->  K  e. SRing )
20 eqidd 2178 . . 3  |-  ( ph  ->  ( .r `  K
)  =  ( .r
`  K ) )
2117, 18, 19, 20dvdsrvald 13215 . 2  |-  ( ph  ->  ( ||r `
 K )  =  { <. y ,  z
>.  |  ( y  e.  ( Base `  K
)  /\  E. x  e.  ( Base `  K
) ( x ( .r `  K ) y )  =  z ) } )
22 eqidd 2178 . . 3  |-  ( ph  ->  ( Base `  L
)  =  ( Base `  L ) )
23 eqidd 2178 . . 3  |-  ( ph  ->  ( ||r `
 L )  =  ( ||r `
 L ) )
24 dvdsrpropdg.l . . 3  |-  ( ph  ->  L  e. SRing )
25 eqidd 2178 . . 3  |-  ( ph  ->  ( .r `  L
)  =  ( .r
`  L ) )
2622, 23, 24, 25dvdsrvald 13215 . 2  |-  ( ph  ->  ( ||r `
 L )  =  { <. y ,  z
>.  |  ( y  e.  ( Base `  L
)  /\  E. x  e.  ( Base `  L
) ( x ( .r `  L ) y )  =  z ) } )
2716, 21, 263eqtr4d 2220 1  |-  ( ph  ->  ( ||r `
 K )  =  ( ||r `
 L ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   {copab 4063   ` cfv 5216  (class class class)co 5874   Basecbs 12456   .rcmulr 12531  SRingcsrg 13099   ||rcdsr 13208
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-iota 5178  df-fun 5218  df-fn 5219  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-mgm 12729  df-sgrp 12762  df-mnd 12772  df-mgp 13084  df-srg 13100  df-dvdsr 13211
This theorem is referenced by:  unitpropdg  13270
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