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Theorem drngprop 17602
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 2455 . . . . . 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 6283 . . . . . . 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 17541 . . . . 5  |-  ( K  e.  Ring  ->  (Unit `  K )  =  (Unit `  L ) )
8 drngprop.p . . . . . . . . . 10  |-  ( +g  `  K )  =  ( +g  `  L )
98oveqi 6283 . . . . . . . . 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 16087 . . . . . . 7  |-  ( K  e.  Ring  ->  ( 0g
`  K )  =  ( 0g `  L
) )
1211sneqd 4028 . . . . . 6  |-  ( K  e.  Ring  ->  { ( 0g `  K ) }  =  { ( 0g `  L ) } )
1312difeq2d 3608 . . . . 5  |-  ( K  e.  Ring  ->  ( (
Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) )
147, 13eqeq12d 2476 . . . 4  |-  ( K  e.  Ring  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
1514pm5.32i 635 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( K  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
162, 8, 4ringprop 17427 . . . 4  |-  ( K  e.  Ring  <->  L  e.  Ring )
1716anbi1i 693 . . 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 2454 . . 3  |-  ( Base `  K )  =  (
Base `  K )
20 eqid 2454 . . 3  |-  (Unit `  K )  =  (Unit `  K )
21 eqid 2454 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2219, 20, 21isdrng 17595 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
23 eqid 2454 . . 3  |-  (Unit `  L )  =  (Unit `  L )
24 eqid 2454 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
252, 23, 24isdrng 17595 . 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 367    = wceq 1398    e. wcel 1823    \ cdif 3458   {csn 4016   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   .rcmulr 14785   0gc0g 14929   Ringcrg 17393  Unitcui 17483   DivRingcdr 17591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-drng 17593
This theorem is referenced by: (None)
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