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Theorem isdrngrd 15538
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 15537 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 2283 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3 eqid 2283 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42, 3opprbas 15411 . . . 4  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
51, 4syl6eq 2331 . . 3  |-  ( ph  ->  B  =  ( Base `  (oppr
`  R ) ) )
6 eqidd 2284 . . 3  |-  ( ph  ->  ( .r `  (oppr `  R
) )  =  ( .r `  (oppr `  R
) ) )
7 isdrngd.z . . . 4  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8 eqid 2283 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
92, 8oppr0 15415 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
107, 9syl6eq 2331 . . 3  |-  ( ph  ->  .0.  =  ( 0g
`  (oppr
`  R ) ) )
11 isdrngd.u . . . 4  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
12 eqid 2283 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
132, 12oppr1 15416 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1411, 13syl6eq 2331 . . 3  |-  ( ph  ->  .1.  =  ( 1r
`  (oppr
`  R ) ) )
15 isdrngd.r . . . 4  |-  ( ph  ->  R  e.  Ring )
162opprrng 15413 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
1715, 16syl 15 . . 3  |-  ( ph  ->  (oppr
`  R )  e. 
Ring )
18 eleq1 2343 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
19 neeq1 2454 . . . . . . 7  |-  ( y  =  x  ->  (
y  =/=  .0.  <->  x  =/=  .0.  ) )
2018, 19anbi12d 691 . . . . . 6  |-  ( y  =  x  ->  (
( y  e.  B  /\  y  =/=  .0.  ) 
<->  ( x  e.  B  /\  x  =/=  .0.  ) ) )
21203anbi2d 1257 . . . . 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 5865 . . . . . 6  |-  ( y  =  x  ->  (
y ( .r `  (oppr `  R ) ) z )  =  ( x ( .r `  (oppr `  R
) ) z ) )
2322neeq1d 2459 . . . . 5  |-  ( y  =  x  ->  (
( y ( .r
`  (oppr
`  R ) ) z )  =/=  .0.  <->  (
x ( .r `  (oppr `  R ) ) z )  =/=  .0.  )
)
2421, 23imbi12d 311 . . . 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 2343 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
26 neeq1 2454 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =/=  .0.  <->  z  =/=  .0.  ) )
2725, 26anbi12d 691 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  B  /\  x  =/=  .0.  ) 
<->  ( z  e.  B  /\  z  =/=  .0.  ) ) )
28273anbi3d 1258 . . . . . 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 5866 . . . . . . 7  |-  ( x  =  z  ->  (
y ( .r `  (oppr `  R ) ) x )  =  ( y ( .r `  (oppr `  R
) ) z ) )
3029neeq1d 2459 . . . . . 6  |-  ( x  =  z  ->  (
( y ( .r
`  (oppr
`  R ) ) x )  =/=  .0.  <->  (
y ( .r `  (oppr `  R ) ) z )  =/=  .0.  )
)
3128, 30imbi12d 311 . . . . 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 976 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
3433oveqd 5875 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =  ( x ( .r `  R ) y ) )
35 eqid 2283 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
36 eqid 2283 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
373, 35, 2, 36opprmul 15408 . . . . . . . 8  |-  ( y ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) y )
3834, 37syl6eqr 2333 . . . . . . 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 2466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
41403com23 1157 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
x  e.  B  /\  x  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
4231, 41chvarv 1953 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) z )  =/=  .0.  )
4324, 42chvarv 1953 . . 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 15408 . . . 4  |-  ( I ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) I )
4832adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
4948oveqd 5875 . . . . 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 2317 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x ( .r
`  R ) I )  =  .1.  )
5247, 51syl5eq 2327 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I ( .r
`  (oppr
`  R ) ) x )  =  .1.  )
535, 6, 10, 14, 17, 43, 44, 45, 46, 52isdrngd 15537 . 2  |-  ( ph  ->  (oppr
`  R )  e.  DivRing )
542opprdrng 15536 . 2  |-  ( R  e.  DivRing 
<->  (oppr
`  R )  e.  DivRing )
5553, 54sylibr 203 1  |-  ( ph  ->  R  e.  DivRing )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209   0gc0g 13400   Ringcrg 15337   1rcur 15339  opprcoppr 15404   DivRingcdr 15512
This theorem is referenced by:  erngdvlem4-rN  31188
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514
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