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Theorem isdrngrd 15587
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 15586 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 2316 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3 eqid 2316 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42, 3opprbas 15460 . . . 4  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
51, 4syl6eq 2364 . . 3  |-  ( ph  ->  B  =  ( Base `  (oppr
`  R ) ) )
6 eqidd 2317 . . 3  |-  ( ph  ->  ( .r `  (oppr `  R
) )  =  ( .r `  (oppr `  R
) ) )
7 isdrngd.z . . . 4  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8 eqid 2316 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
92, 8oppr0 15464 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
107, 9syl6eq 2364 . . 3  |-  ( ph  ->  .0.  =  ( 0g
`  (oppr
`  R ) ) )
11 isdrngd.u . . . 4  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
12 eqid 2316 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
132, 12oppr1 15465 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1411, 13syl6eq 2364 . . 3  |-  ( ph  ->  .1.  =  ( 1r
`  (oppr
`  R ) ) )
15 isdrngd.r . . . 4  |-  ( ph  ->  R  e.  Ring )
162opprrng 15462 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
1715, 16syl 15 . . 3  |-  ( ph  ->  (oppr
`  R )  e. 
Ring )
18 eleq1 2376 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
19 neeq1 2487 . . . . . . 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 5907 . . . . . 6  |-  ( y  =  x  ->  (
y ( .r `  (oppr `  R ) ) z )  =  ( x ( .r `  (oppr `  R
) ) z ) )
2322neeq1d 2492 . . . . 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 2376 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
26 neeq1 2487 . . . . . . . 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 5908 . . . . . . 7  |-  ( x  =  z  ->  (
y ( .r `  (oppr `  R ) ) x )  =  ( y ( .r `  (oppr `  R
) ) z ) )
3029neeq1d 2492 . . . . . 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 5917 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =  ( x ( .r `  R ) y ) )
35 eqid 2316 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
36 eqid 2316 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
373, 35, 2, 36opprmul 15457 . . . . . . . 8  |-  ( y ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) y )
3834, 37syl6eqr 2366 . . . . . . 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 2499 . . . . . 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 1985 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) z )  =/=  .0.  )
4324, 42chvarv 1985 . . 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 15457 . . . 4  |-  ( I ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) I )
4832adantr 451 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
4948oveqd 5917 . . . . 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 2350 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x ( .r
`  R ) I )  =  .1.  )
5247, 51syl5eq 2360 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I ( .r
`  (oppr
`  R ) ) x )  =  .1.  )
535, 6, 10, 14, 17, 43, 44, 45, 46, 52isdrngd 15586 . 2  |-  ( ph  ->  (oppr
`  R )  e.  DivRing )
542opprdrng 15585 . 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 1633    e. wcel 1701    =/= wne 2479   ` cfv 5292  (class class class)co 5900   Basecbs 13195   .rcmulr 13256   0gc0g 13449   Ringcrg 15386   1rcur 15388  opprcoppr 15453   DivRingcdr 15561
This theorem is referenced by:  erngdvlem4-rN  31006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-tpos 6276  df-riota 6346  df-recs 6430  df-rdg 6465  df-er 6702  df-en 6907  df-dom 6908  df-sdom 6909  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-nn 9792  df-2 9849  df-3 9850  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-0g 13453  df-mnd 14416  df-grp 14538  df-minusg 14539  df-mgp 15375  df-rng 15389  df-ur 15391  df-oppr 15454  df-dvdsr 15472  df-unit 15473  df-invr 15503  df-dvr 15514  df-drng 15563
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