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Theorem isdrngrd 16856
Description: Properties that determine a division ring.  I (reciprocal) is normally dependent on  x i.e. read it as  I ( x )." This version of isdrngd 16855 requires a right reciprocal instead of left. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
isdrngd.b  |-  ( ph  ->  B  =  ( Base `  R ) )
isdrngd.t  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
isdrngd.z  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
isdrngd.u  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
isdrngd.r  |-  ( ph  ->  R  e.  Ring )
isdrngd.n  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =/=  .0.  )
isdrngd.o  |-  ( ph  ->  .1.  =/=  .0.  )
isdrngd.i  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  B )
isdrngd.j  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  =/=  .0.  )
isdrngrd.k  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x  .x.  I
)  =  .1.  )
Assertion
Ref Expression
isdrngrd  |-  ( ph  ->  R  e.  DivRing )
Distinct variable groups:    x, y,  .0.    x,  .1. , y    x, B, y    y, I    x, R, y    ph, x, y   
x,  .x. , y
Allowed substitution hint:    I( x)

Proof of Theorem isdrngrd
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 isdrngd.b . . . 4  |-  ( ph  ->  B  =  ( Base `  R ) )
2 eqid 2441 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3 eqid 2441 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42, 3opprbas 16719 . . . 4  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
51, 4syl6eq 2489 . . 3  |-  ( ph  ->  B  =  ( Base `  (oppr
`  R ) ) )
6 eqidd 2442 . . 3  |-  ( ph  ->  ( .r `  (oppr `  R
) )  =  ( .r `  (oppr `  R
) ) )
7 isdrngd.z . . . 4  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8 eqid 2441 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
92, 8oppr0 16723 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
107, 9syl6eq 2489 . . 3  |-  ( ph  ->  .0.  =  ( 0g
`  (oppr
`  R ) ) )
11 isdrngd.u . . . 4  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
12 eqid 2441 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
132, 12oppr1 16724 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1411, 13syl6eq 2489 . . 3  |-  ( ph  ->  .1.  =  ( 1r
`  (oppr
`  R ) ) )
15 isdrngd.r . . . 4  |-  ( ph  ->  R  e.  Ring )
162opprrng 16721 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
1715, 16syl 16 . . 3  |-  ( ph  ->  (oppr
`  R )  e. 
Ring )
18 eleq1 2501 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
19 neeq1 2614 . . . . . . 7  |-  ( y  =  x  ->  (
y  =/=  .0.  <->  x  =/=  .0.  ) )
2018, 19anbi12d 710 . . . . . 6  |-  ( y  =  x  ->  (
( y  e.  B  /\  y  =/=  .0.  ) 
<->  ( x  e.  B  /\  x  =/=  .0.  ) ) )
21203anbi2d 1294 . . . . 5  |-  ( y  =  x  ->  (
( ph  /\  (
y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/=  .0.  ) )  <->  ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/= 
.0.  ) ) ) )
22 oveq1 6096 . . . . . 6  |-  ( y  =  x  ->  (
y ( .r `  (oppr `  R ) ) z )  =  ( x ( .r `  (oppr `  R
) ) z ) )
2322neeq1d 2619 . . . . 5  |-  ( y  =  x  ->  (
( y ( .r
`  (oppr
`  R ) ) z )  =/=  .0.  <->  (
x ( .r `  (oppr `  R ) ) z )  =/=  .0.  )
)
2421, 23imbi12d 320 . . . 4  |-  ( y  =  x  ->  (
( ( ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/= 
.0.  ) )  -> 
( y ( .r
`  (oppr
`  R ) ) z )  =/=  .0.  ) 
<->  ( ( ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/= 
.0.  ) )  -> 
( x ( .r
`  (oppr
`  R ) ) z )  =/=  .0.  ) ) )
25 eleq1 2501 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
26 neeq1 2614 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =/=  .0.  <->  z  =/=  .0.  ) )
2725, 26anbi12d 710 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  B  /\  x  =/=  .0.  ) 
<->  ( z  e.  B  /\  z  =/=  .0.  ) ) )
28273anbi3d 1295 . . . . . 6  |-  ( x  =  z  ->  (
( ph  /\  (
y  e.  B  /\  y  =/=  .0.  )  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  <->  ( ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/= 
.0.  ) ) ) )
29 oveq2 6097 . . . . . . 7  |-  ( x  =  z  ->  (
y ( .r `  (oppr `  R ) ) x )  =  ( y ( .r `  (oppr `  R
) ) z ) )
3029neeq1d 2619 . . . . . 6  |-  ( x  =  z  ->  (
( y ( .r
`  (oppr
`  R ) ) x )  =/=  .0.  <->  (
y ( .r `  (oppr `  R ) ) z )  =/=  .0.  )
)
3128, 30imbi12d 320 . . . . 5  |-  ( x  =  z  ->  (
( ( ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  ( x  e.  B  /\  x  =/= 
.0.  ) )  -> 
( y ( .r
`  (oppr
`  R ) ) x )  =/=  .0.  ) 
<->  ( ( ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  ( z  e.  B  /\  z  =/= 
.0.  ) )  -> 
( y ( .r
`  (oppr
`  R ) ) z )  =/=  .0.  ) ) )
32 isdrngd.t . . . . . . . . . 10  |-  ( ph  ->  .x.  =  ( .r
`  R ) )
33323ad2ant1 1009 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
3433oveqd 6106 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =  ( x ( .r `  R ) y ) )
35 eqid 2441 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
36 eqid 2441 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
373, 35, 2, 36opprmul 16716 . . . . . . . 8  |-  ( y ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) y )
3834, 37syl6eqr 2491 . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =  ( y ( .r `  (oppr `  R ) ) x ) )
39 isdrngd.n . . . . . . 7  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =/=  .0.  )
4038, 39eqnetrrd 2626 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
41403com23 1193 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
x  e.  B  /\  x  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
4231, 41chvarv 1958 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) z )  =/=  .0.  )
4324, 42chvarv 1958 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( x ( .r `  (oppr `  R
) ) z )  =/=  .0.  )
44 isdrngd.o . . 3  |-  ( ph  ->  .1.  =/=  .0.  )
45 isdrngd.i . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  e.  B )
46 isdrngd.j . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  I  =/=  .0.  )
473, 35, 2, 36opprmul 16716 . . . 4  |-  ( I ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) I )
4832adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
4948oveqd 6106 . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x  .x.  I
)  =  ( x ( .r `  R
) I ) )
50 isdrngrd.k . . . . 5  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x  .x.  I
)  =  .1.  )
5149, 50eqtr3d 2475 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x ( .r
`  R ) I )  =  .1.  )
5247, 51syl5eq 2485 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I ( .r
`  (oppr
`  R ) ) x )  =  .1.  )
535, 6, 10, 14, 17, 43, 44, 45, 46, 52isdrngd 16855 . 2  |-  ( ph  ->  (oppr
`  R )  e.  DivRing )
542opprdrng 16854 . 2  |-  ( R  e.  DivRing 
<->  (oppr
`  R )  e.  DivRing )
5553, 54sylibr 212 1  |-  ( ph  ->  R  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2604   ` cfv 5416  (class class class)co 6089   Basecbs 14172   .rcmulr 14237   0gc0g 14376   1rcur 16601   Ringcrg 16643  opprcoppr 16712   DivRingcdr 16830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-tpos 6743  df-recs 6830  df-rdg 6864  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-2 10378  df-3 10379  df-ndx 14175  df-slot 14176  df-base 14177  df-sets 14178  df-ress 14179  df-plusg 14249  df-mulr 14250  df-0g 14378  df-mnd 15413  df-grp 15543  df-minusg 15544  df-mgp 16590  df-ur 16602  df-rng 16645  df-oppr 16713  df-dvdsr 16731  df-unit 16732  df-invr 16762  df-dvr 16773  df-drng 16832
This theorem is referenced by:  erngdvlem4-rN  34640
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