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Theorem drngpropd 15555
Description: If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.)
Hypotheses
Ref Expression
drngpropd.1  |-  ( ph  ->  B  =  ( Base `  K ) )
drngpropd.2  |-  ( ph  ->  B  =  ( Base `  L ) )
drngpropd.3  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
drngpropd.4  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
Assertion
Ref Expression
drngpropd  |-  ( ph  ->  ( K  e.  DivRing  <->  L  e.  DivRing ) )
Distinct variable groups:    x, y, B    x, K, y    ph, x, y    x, L, y

Proof of Theorem drngpropd
StepHypRef Expression
1 drngpropd.1 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  K ) )
2 drngpropd.2 . . . . . . 7  |-  ( ph  ->  B  =  ( Base `  L ) )
3 drngpropd.4 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
41, 2, 3unitpropd 15495 . . . . . 6  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
54adantr 451 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  (Unit `  K
)  =  (Unit `  L ) )
61, 2eqtr3d 2330 . . . . . . 7  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
76adantr 451 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  ( Base `  K )  =  (
Base `  L )
)
81adantr 451 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  K )
)
92adantr 451 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  L )
)
10 drngpropd.3 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  y  e.  B ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
1110adantlr 695 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
128, 9, 11grpidpropd 14415 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( 0g `  K )  =  ( 0g `  L ) )
1312sneqd 3666 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  { ( 0g `  K ) }  =  { ( 0g
`  L ) } )
147, 13difeq12d 3308 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  ( ( Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) )
155, 14eqeq12d 2310 . . . 4  |-  ( (
ph  /\  K  e.  Ring )  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) ) )
1615pm5.32da 622 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  K
)  =  ( (
Base `  K )  \  { ( 0g `  K ) } ) )  <->  ( K  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
171, 2, 10, 3rngpropd 15388 . . . 4  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
1817anbi1d 685 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) )  <->  ( L  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
1916, 18bitrd 244 . 2  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  K
)  =  ( (
Base `  K )  \  { ( 0g `  K ) } ) )  <->  ( L  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
20 eqid 2296 . . 3  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2296 . . 3  |-  (Unit `  K )  =  (Unit `  K )
22 eqid 2296 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2320, 21, 22isdrng 15532 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
24 eqid 2296 . . 3  |-  ( Base `  L )  =  (
Base `  L )
25 eqid 2296 . . 3  |-  (Unit `  L )  =  (Unit `  L )
26 eqid 2296 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2724, 25, 26isdrng 15532 . 2  |-  ( L  e.  DivRing 
<->  ( L  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) ) )
2819, 23, 273bitr4g 279 1  |-  ( ph  ->  ( K  e.  DivRing  <->  L  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    \ cdif 3162   {csn 3653   ` cfv 5271  (class class class)co 5874   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   0gc0g 13416   Ringcrg 15353  Unitcui 15437   DivRingcdr 15528
This theorem is referenced by:  fldpropd  15556  lvecprop2d  15935  hlhildrng  32767
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-drng 15530
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