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Theorem drngprop 15616
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 2359 . . . . . 6  |-  ( K  e.  Ring  ->  ( Base `  K )  =  (
Base `  K )
)
2 drngprop.b . . . . . . 7  |-  ( Base `  K )  =  (
Base `  L )
32a1i 10 . . . . . 6  |-  ( K  e.  Ring  ->  ( Base `  K )  =  (
Base `  L )
)
4 drngprop.m . . . . . . . 8  |-  ( .r
`  K )  =  ( .r `  L
)
54oveqi 5955 . . . . . . 7  |-  ( x ( .r `  K
) y )  =  ( x ( .r
`  L ) y )
65a1i 10 . . . . . 6  |-  ( ( K  e.  Ring  /\  (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) ) )  -> 
( x ( .r
`  K ) y )  =  ( x ( .r `  L
) y ) )
71, 3, 6unitpropd 15572 . . . . 5  |-  ( K  e.  Ring  ->  (Unit `  K )  =  (Unit `  L ) )
8 drngprop.p . . . . . . . . . 10  |-  ( +g  `  K )  =  ( +g  `  L )
98oveqi 5955 . . . . . . . . 9  |-  ( x ( +g  `  K
) y )  =  ( x ( +g  `  L ) y )
109a1i 10 . . . . . . . 8  |-  ( ( K  e.  Ring  /\  (
x  e.  ( Base `  K )  /\  y  e.  ( Base `  K
) ) )  -> 
( x ( +g  `  K ) y )  =  ( x ( +g  `  L ) y ) )
111, 3, 10grpidpropd 14492 . . . . . . 7  |-  ( K  e.  Ring  ->  ( 0g
`  K )  =  ( 0g `  L
) )
1211sneqd 3729 . . . . . 6  |-  ( K  e.  Ring  ->  { ( 0g `  K ) }  =  { ( 0g `  L ) } )
1312difeq2d 3370 . . . . 5  |-  ( K  e.  Ring  ->  ( (
Base `  K )  \  { ( 0g `  K ) } )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) )
147, 13eqeq12d 2372 . . . 4  |-  ( K  e.  Ring  ->  ( (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } )  <->  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
1514pm5.32i 618 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( K  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
162, 8, 4rngprop 15467 . . . 4  |-  ( K  e.  Ring  <->  L  e.  Ring )
1716anbi1i 676 . . 3  |-  ( ( K  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  K
)  \  { ( 0g `  L ) } ) )  <->  ( L  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
1815, 17bitri 240 . 2  |-  ( ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K
)  \  { ( 0g `  K ) } ) )  <->  ( L  e.  Ring  /\  (Unit `  L
)  =  ( (
Base `  K )  \  { ( 0g `  L ) } ) ) )
19 eqid 2358 . . 3  |-  ( Base `  K )  =  (
Base `  K )
20 eqid 2358 . . 3  |-  (Unit `  K )  =  (Unit `  K )
21 eqid 2358 . . 3  |-  ( 0g
`  K )  =  ( 0g `  K
)
2219, 20, 21isdrng 15609 . 2  |-  ( K  e.  DivRing 
<->  ( K  e.  Ring  /\  (Unit `  K )  =  ( ( Base `  K )  \  {
( 0g `  K
) } ) ) )
23 eqid 2358 . . 3  |-  (Unit `  L )  =  (Unit `  L )
24 eqid 2358 . . 3  |-  ( 0g
`  L )  =  ( 0g `  L
)
252, 23, 24isdrng 15609 . 2  |-  ( L  e.  DivRing 
<->  ( L  e.  Ring  /\  (Unit `  L )  =  ( ( Base `  K )  \  {
( 0g `  L
) } ) ) )
2618, 22, 253bitr4i 268 1  |-  ( K  e.  DivRing 
<->  L  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1642    e. wcel 1710    \ cdif 3225   {csn 3716   ` cfv 5334  (class class class)co 5942   Basecbs 13239   +g cplusg 13299   .rcmulr 13300   0gc0g 13493   Ringcrg 15430  Unitcui 15514   DivRingcdr 15605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591  ax-cnex 8880  ax-resscn 8881  ax-1cn 8882  ax-icn 8883  ax-addcl 8884  ax-addrcl 8885  ax-mulcl 8886  ax-mulrcl 8887  ax-mulcom 8888  ax-addass 8889  ax-mulass 8890  ax-distr 8891  ax-i2m1 8892  ax-1ne0 8893  ax-1rid 8894  ax-rnegex 8895  ax-rrecex 8896  ax-cnre 8897  ax-pre-lttri 8898  ax-pre-lttrn 8899  ax-pre-ltadd 8900  ax-pre-mulgt0 8901
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-lim 4476  df-suc 4477  df-om 4736  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-tpos 6318  df-riota 6388  df-recs 6472  df-rdg 6507  df-er 6744  df-en 6949  df-dom 6950  df-sdom 6951  df-pnf 8956  df-mnf 8957  df-xr 8958  df-ltxr 8959  df-le 8960  df-sub 9126  df-neg 9127  df-nn 9834  df-2 9891  df-3 9892  df-ndx 13242  df-slot 13243  df-base 13244  df-sets 13245  df-plusg 13312  df-mulr 13313  df-0g 13497  df-mnd 14460  df-grp 14582  df-mgp 15419  df-rng 15433  df-ur 15435  df-oppr 15498  df-dvdsr 15516  df-unit 15517  df-drng 15607
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