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Theorem isdrngrd 17222
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 17221 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 2467 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3 eqid 2467 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42, 3opprbas 17079 . . . 4  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
51, 4syl6eq 2524 . . 3  |-  ( ph  ->  B  =  ( Base `  (oppr
`  R ) ) )
6 eqidd 2468 . . 3  |-  ( ph  ->  ( .r `  (oppr `  R
) )  =  ( .r `  (oppr `  R
) ) )
7 isdrngd.z . . . 4  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8 eqid 2467 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
92, 8oppr0 17083 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
107, 9syl6eq 2524 . . 3  |-  ( ph  ->  .0.  =  ( 0g
`  (oppr
`  R ) ) )
11 isdrngd.u . . . 4  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
12 eqid 2467 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
132, 12oppr1 17084 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1411, 13syl6eq 2524 . . 3  |-  ( ph  ->  .1.  =  ( 1r
`  (oppr
`  R ) ) )
15 isdrngd.r . . . 4  |-  ( ph  ->  R  e.  Ring )
162opprrng 17081 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
1715, 16syl 16 . . 3  |-  ( ph  ->  (oppr
`  R )  e. 
Ring )
18 eleq1 2539 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
19 neeq1 2748 . . . . . . 7  |-  ( y  =  x  ->  (
y  =/=  .0.  <->  x  =/=  .0.  ) )
2018, 19anbi12d 710 . . . . . 6  |-  ( y  =  x  ->  (
( y  e.  B  /\  y  =/=  .0.  ) 
<->  ( x  e.  B  /\  x  =/=  .0.  ) ) )
21203anbi2d 1304 . . . . 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 6291 . . . . . 6  |-  ( y  =  x  ->  (
y ( .r `  (oppr `  R ) ) z )  =  ( x ( .r `  (oppr `  R
) ) z ) )
2322neeq1d 2744 . . . . 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 2539 . . . . . . . 8  |-  ( x  =  z  ->  (
x  e.  B  <->  z  e.  B ) )
26 neeq1 2748 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =/=  .0.  <->  z  =/=  .0.  ) )
2725, 26anbi12d 710 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  B  /\  x  =/=  .0.  ) 
<->  ( z  e.  B  /\  z  =/=  .0.  ) ) )
28273anbi3d 1305 . . . . . 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 6292 . . . . . . 7  |-  ( x  =  z  ->  (
y ( .r `  (oppr `  R ) ) x )  =  ( y ( .r `  (oppr `  R
) ) z ) )
3029neeq1d 2744 . . . . . 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 1017 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
3433oveqd 6301 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =  ( x ( .r `  R ) y ) )
35 eqid 2467 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
36 eqid 2467 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
373, 35, 2, 36opprmul 17076 . . . . . . . 8  |-  ( y ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) y )
3834, 37syl6eqr 2526 . . . . . . 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 2761 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
41403com23 1202 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
x  e.  B  /\  x  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
4231, 41chvarv 1983 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) z )  =/=  .0.  )
4324, 42chvarv 1983 . . 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 17076 . . . 4  |-  ( I ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) I )
4832adantr 465 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
4948oveqd 6301 . . . . 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 2510 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x ( .r
`  R ) I )  =  .1.  )
5247, 51syl5eq 2520 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I ( .r
`  (oppr
`  R ) ) x )  =  .1.  )
535, 6, 10, 14, 17, 43, 44, 45, 46, 52isdrngd 17221 . 2  |-  ( ph  ->  (oppr
`  R )  e.  DivRing )
542opprdrng 17220 . 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5588  (class class class)co 6284   Basecbs 14490   .rcmulr 14556   0gc0g 14695   1rcur 16955   Ringcrg 17000  opprcoppr 17072   DivRingcdr 17196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-tpos 6955  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-2 10594  df-3 10595  df-ndx 14493  df-slot 14494  df-base 14495  df-sets 14496  df-ress 14497  df-plusg 14568  df-mulr 14569  df-0g 14697  df-mnd 15732  df-grp 15867  df-minusg 15868  df-mgp 16944  df-ur 16956  df-rng 17002  df-oppr 17073  df-dvdsr 17091  df-unit 17092  df-invr 17122  df-dvr 17133  df-drng 17198
This theorem is referenced by:  erngdvlem4-rN  35813
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