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Theorem drngpropd 15863
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 15803 . . . . . 6  |-  ( ph  ->  (Unit `  K )  =  (Unit `  L )
)
54adantr 453 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  (Unit `  K
)  =  (Unit `  L ) )
61, 2eqtr3d 2471 . . . . . . 7  |-  ( ph  ->  ( Base `  K
)  =  ( Base `  L ) )
76adantr 453 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  ( Base `  K )  =  (
Base `  L )
)
81adantr 453 . . . . . . . 8  |-  ( (
ph  /\  K  e.  Ring )  ->  B  =  ( Base `  K )
)
92adantr 453 . . . . . . . 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 697 . . . . . . . 8  |-  ( ( ( ph  /\  K  e.  Ring )  /\  (
x  e.  B  /\  y  e.  B )
)  ->  ( x
( +g  `  K ) y )  =  ( x ( +g  `  L
) y ) )
128, 9, 11grpidpropd 14723 . . . . . . 7  |-  ( (
ph  /\  K  e.  Ring )  ->  ( 0g `  K )  =  ( 0g `  L ) )
1312sneqd 3828 . . . . . 6  |-  ( (
ph  /\  K  e.  Ring )  ->  { ( 0g `  K ) }  =  { ( 0g
`  L ) } )
147, 13difeq12d 3467 . . . . 5  |-  ( (
ph  /\  K  e.  Ring )  ->  ( ( Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) )
155, 14eqeq12d 2451 . . . 4  |-  ( (
ph  /\  K  e.  Ring )  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) ) )
1615pm5.32da 624 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  K
)  =  ( (
Base `  K )  \  { ( 0g `  K ) } ) )  <->  ( K  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
171, 2, 10, 3rngpropd 15696 . . . 4  |-  ( ph  ->  ( K  e.  Ring  <->  L  e.  Ring ) )
1817anbi1d 687 . . 3  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) )  <->  ( L  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
1916, 18bitrd 246 . 2  |-  ( ph  ->  ( ( K  e. 
Ring  /\  (Unit `  K
)  =  ( (
Base `  K )  \  { ( 0g `  K ) } ) )  <->  ( L  e. 
Ring  /\  (Unit `  L
)  =  ( (
Base `  L )  \  { ( 0g `  L ) } ) ) ) )
20 eqid 2437 . . 3  |-  ( Base `  K )  =  (
Base `  K )
21 eqid 2437 . . 3  |-  (Unit `  K )  =  (Unit `  K )
22 eqid 2437 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2320, 21, 22isdrng 15840 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
24 eqid 2437 . . 3  |-  ( Base `  L )  =  (
Base `  L )
25 eqid 2437 . . 3  |-  (Unit `  L )  =  (Unit `  L )
26 eqid 2437 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
2724, 25, 26isdrng 15840 . 2  |-  ( L  e.  DivRing 
<->  ( L  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  L )  \  {
( 0g `  L
) } ) ) )
2819, 23, 273bitr4g 281 1  |-  ( ph  ->  ( K  e.  DivRing  <->  L  e.  DivRing ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    \ cdif 3318   {csn 3815   ` cfv 5455  (class class class)co 6082   Basecbs 13470   +g cplusg 13530   .rcmulr 13531   0gc0g 13724   Ringcrg 15661  Unitcui 15745   DivRingcdr 15836
This theorem is referenced by:  fldpropd  15864  lvecprop2d  16239  hlhildrng  32754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-tpos 6480  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-nn 10002  df-2 10059  df-3 10060  df-ndx 13473  df-slot 13474  df-base 13475  df-sets 13476  df-plusg 13543  df-mulr 13544  df-0g 13728  df-mnd 14691  df-grp 14813  df-mgp 15650  df-rng 15664  df-ur 15666  df-oppr 15729  df-dvdsr 15747  df-unit 15748  df-drng 15838
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