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Theorem drngprop 16946
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 2452 . . . . . 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 6200 . . . . . . 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 16892 . . . . 5  |-  ( K  e.  Ring  ->  (Unit `  K )  =  (Unit `  L ) )
8 drngprop.p . . . . . . . . . 10  |-  ( +g  `  K )  =  ( +g  `  L )
98oveqi 6200 . . . . . . . . 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 15546 . . . . . . 7  |-  ( K  e.  Ring  ->  ( 0g
`  K )  =  ( 0g `  L
) )
1211sneqd 3984 . . . . . 6  |-  ( K  e.  Ring  ->  { ( 0g `  K ) }  =  { ( 0g `  L ) } )
1312difeq2d 3569 . . . . 5  |-  ( K  e.  Ring  ->  ( (
Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) )
147, 13eqeq12d 2472 . . . 4  |-  ( K  e.  Ring  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
1514pm5.32i 637 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( K  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
162, 8, 4rngprop 16781 . . . 4  |-  ( K  e.  Ring  <->  L  e.  Ring )
1716anbi1i 695 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  K
)  \  { ( 0g `  L ) } ) )  <->  ( L  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
1815, 17bitri 249 . 2  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( L  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
19 eqid 2451 . . 3  |-  ( Base `  K )  =  (
Base `  K )
20 eqid 2451 . . 3  |-  (Unit `  K )  =  (Unit `  K )
21 eqid 2451 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2219, 20, 21isdrng 16939 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
23 eqid 2451 . . 3  |-  (Unit `  L )  =  (Unit `  L )
24 eqid 2451 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
252, 23, 24isdrng 16939 . 2  |-  ( L  e.  DivRing 
<->  ( L  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
2618, 22, 253bitr4i 277 1  |-  ( K  e.  DivRing 
<->  L  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3420   {csn 3972   ` cfv 5513  (class class class)co 6187   Basecbs 14273   +g cplusg 14337   .rcmulr 14338   0gc0g 14477   Ringcrg 16748  Unitcui 16834   DivRingcdr 16935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469  ax-cnex 9436  ax-resscn 9437  ax-1cn 9438  ax-icn 9439  ax-addcl 9440  ax-addrcl 9441  ax-mulcl 9442  ax-mulrcl 9443  ax-mulcom 9444  ax-addass 9445  ax-mulass 9446  ax-distr 9447  ax-i2m1 9448  ax-1ne0 9449  ax-1rid 9450  ax-rnegex 9451  ax-rrecex 9452  ax-cnre 9453  ax-pre-lttri 9454  ax-pre-lttrn 9455  ax-pre-ltadd 9456  ax-pre-mulgt0 9457
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-nel 2645  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-pss 3439  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4187  df-iun 4268  df-br 4388  df-opab 4446  df-mpt 4447  df-tr 4481  df-eprel 4727  df-id 4731  df-po 4736  df-so 4737  df-fr 4774  df-we 4776  df-ord 4817  df-on 4818  df-lim 4819  df-suc 4820  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-riota 6148  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-om 6574  df-tpos 6842  df-recs 6929  df-rdg 6963  df-er 7198  df-en 7408  df-dom 7409  df-sdom 7410  df-pnf 9518  df-mnf 9519  df-xr 9520  df-ltxr 9521  df-le 9522  df-sub 9695  df-neg 9696  df-nn 10421  df-2 10478  df-3 10479  df-ndx 14276  df-slot 14277  df-base 14278  df-sets 14279  df-plusg 14350  df-mulr 14351  df-0g 14479  df-mnd 15514  df-grp 15644  df-mgp 16694  df-ur 16706  df-rng 16750  df-oppr 16818  df-dvdsr 16836  df-unit 16837  df-drng 16937
This theorem is referenced by: (None)
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