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Theorem drngprop 15809
Description: If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.)
Hypotheses
Ref Expression
drngprop.b  |-  ( Base `  K )  =  (
Base `  L )
drngprop.p  |-  ( +g  `  K )  =  ( +g  `  L )
drngprop.m  |-  ( .r
`  K )  =  ( .r `  L
)
Assertion
Ref Expression
drngprop  |-  ( K  e.  DivRing 
<->  L  e.  DivRing )

Proof of Theorem drngprop
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2413 . . . . . 6  |-  ( K  e.  Ring  ->  ( Base `  K )  =  (
Base `  K )
)
2 drngprop.b . . . . . . 7  |-  ( Base `  K )  =  (
Base `  L )
32a1i 11 . . . . . 6  |-  ( K  e.  Ring  ->  ( Base `  K )  =  (
Base `  L )
)
4 drngprop.m . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  L
)
54oveqi 6061 . . . . . . 7  |-  ( x ( .r `  K
) y )  =  ( x ( .r
`  L ) y )
65a1i 11 . . . . . 6  |-  ( ( K  e.  Ring  /\  (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
71, 3, 6unitpropd 15765 . . . . 5  |-  ( K  e.  Ring  ->  (Unit `  K )  =  (Unit `  L ) )
8 drngprop.p . . . . . . . . . 10  |-  ( +g  `  K )  =  ( +g  `  L )
98oveqi 6061 . . . . . . . . 9  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y )
109a1i 11 . . . . . . . 8  |-  ( ( K  e.  Ring  /\  (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
111, 3, 10grpidpropd 14685 . . . . . . 7  |-  ( K  e.  Ring  ->  ( 0g
`  K )  =  ( 0g `  L
) )
1211sneqd 3795 . . . . . 6  |-  ( K  e.  Ring  ->  { ( 0g `  K ) }  =  { ( 0g `  L ) } )
1312difeq2d 3433 . . . . 5  |-  ( K  e.  Ring  ->  ( (
Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) )
147, 13eqeq12d 2426 . . . 4  |-  ( K  e.  Ring  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
1514pm5.32i 619 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( K  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
162, 8, 4rngprop 15660 . . . 4  |-  ( K  e.  Ring  <->  L  e.  Ring )
1716anbi1i 677 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  K
)  \  { ( 0g `  L ) } ) )  <->  ( L  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
1815, 17bitri 241 . 2  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( L  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
19 eqid 2412 . . 3  |-  ( Base `  K )  =  (
Base `  K )
20 eqid 2412 . . 3  |-  (Unit `  K )  =  (Unit `  K )
21 eqid 2412 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2219, 20, 21isdrng 15802 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
23 eqid 2412 . . 3  |-  (Unit `  L )  =  (Unit `  L )
24 eqid 2412 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
252, 23, 24isdrng 15802 . 2  |-  ( L  e.  DivRing 
<->  ( L  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
2618, 22, 253bitr4i 269 1  |-  ( K  e.  DivRing 
<->  L  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721    \ cdif 3285   {csn 3782   ` cfv 5421  (class class class)co 6048   Basecbs 13432   +g cplusg 13492   .rcmulr 13493   0gc0g 13686   Ringcrg 15623  Unitcui 15707   DivRingcdr 15798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-tpos 6446  df-riota 6516  df-recs 6600  df-rdg 6635  df-er 6872  df-en 7077  df-dom 7078  df-sdom 7079  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-nn 9965  df-2 10022  df-3 10023  df-ndx 13435  df-slot 13436  df-base 13437  df-sets 13438  df-plusg 13505  df-mulr 13506  df-0g 13690  df-mnd 14653  df-grp 14775  df-mgp 15612  df-rng 15626  df-ur 15628  df-oppr 15691  df-dvdsr 15709  df-unit 15710  df-drng 15800
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