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Theorem drngpropd 16967
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 16897 . . . . . 6  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
54adantr 465 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  (Unit `  K
)  =  (Unit `  L ) )
61, 2eqtr3d 2494 . . . . . . 7  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
76adantr 465 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  ( Base `  K )  =  (
Base `  L )
)
81adantr 465 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  K )
)
92adantr 465 . . . . . . . 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 714 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
128, 9, 11grpidpropd 15551 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( 0g `  K )  =  ( 0g `  L ) )
1312sneqd 3989 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  { ( 0g `  K ) }  =  { ( 0g
`  L ) } )
147, 13difeq12d 3575 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  ( ( Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) )
155, 14eqeq12d 2473 . . . 4  |-  ( (
ph  /\  K  e.  Ring )  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) ) )
1615pm5.32da 641 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  K
)  =  ( (
Base `  K )  \  { ( 0g `  K ) } ) )  <->  ( K  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
171, 2, 10, 3rngpropd 16784 . . . 4  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
1817anbi1d 704 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) )  <->  ( L  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
1916, 18bitrd 253 . 2  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  K
)  =  ( (
Base `  K )  \  { ( 0g `  K ) } ) )  <->  ( L  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
20 eqid 2451 . . 3  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2451 . . 3  |-  (Unit `  K )  =  (Unit `  K )
22 eqid 2451 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2320, 21, 22isdrng 16944 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
24 eqid 2451 . . 3  |-  ( Base `  L )  =  (
Base `  L )
25 eqid 2451 . . 3  |-  (Unit `  L )  =  (Unit `  L )
26 eqid 2451 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2724, 25, 26isdrng 16944 . 2  |-  ( L  e.  DivRing 
<->  ( L  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) ) )
2819, 23, 273bitr4g 288 1  |-  ( ph  ->  ( K  e.  DivRing  <->  L  e.  DivRing ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3425   {csn 3977   ` cfv 5518  (class class class)co 6192   Basecbs 14278   +g cplusg 14342   .rcmulr 14343   0gc0g 14482   Ringcrg 16753  Unitcui 16839   DivRingcdr 16940
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 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474  ax-cnex 9441  ax-resscn 9442  ax-1cn 9443  ax-icn 9444  ax-addcl 9445  ax-addrcl 9446  ax-mulcl 9447  ax-mulrcl 9448  ax-mulcom 9449  ax-addass 9450  ax-mulass 9451  ax-distr 9452  ax-i2m1 9453  ax-1ne0 9454  ax-1rid 9455  ax-rnegex 9456  ax-rrecex 9457  ax-cnre 9458  ax-pre-lttri 9459  ax-pre-lttrn 9460  ax-pre-ltadd 9461  ax-pre-mulgt0 9462
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 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-pss 3444  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-tp 3982  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4486  df-eprel 4732  df-id 4736  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-riota 6153  df-ov 6195  df-oprab 6196  df-mpt2 6197  df-om 6579  df-tpos 6847  df-recs 6934  df-rdg 6968  df-er 7203  df-en 7413  df-dom 7414  df-sdom 7415  df-pnf 9523  df-mnf 9524  df-xr 9525  df-ltxr 9526  df-le 9527  df-sub 9700  df-neg 9701  df-nn 10426  df-2 10483  df-3 10484  df-ndx 14281  df-slot 14282  df-base 14283  df-sets 14284  df-plusg 14355  df-mulr 14356  df-0g 14484  df-mnd 15519  df-grp 15649  df-mgp 16699  df-ur 16711  df-rng 16755  df-oppr 16823  df-dvdsr 16841  df-unit 16842  df-drng 16942
This theorem is referenced by:  fldpropd  16968  lvecprop2d  17355  hlhildrng  35908
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