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Theorem isdrngrd 17936
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 17935 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 2429 . . . . 5  |-  (oppr `  R
)  =  (oppr `  R
)
3 eqid 2429 . . . . 5  |-  ( Base `  R )  =  (
Base `  R )
42, 3opprbas 17792 . . . 4  |-  ( Base `  R )  =  (
Base `  (oppr
`  R ) )
51, 4syl6eq 2486 . . 3  |-  ( ph  ->  B  =  ( Base `  (oppr
`  R ) ) )
6 eqidd 2430 . . 3  |-  ( ph  ->  ( .r `  (oppr `  R
) )  =  ( .r `  (oppr `  R
) ) )
7 isdrngd.z . . . 4  |-  ( ph  ->  .0.  =  ( 0g
`  R ) )
8 eqid 2429 . . . . 5  |-  ( 0g
`  R )  =  ( 0g `  R
)
92, 8oppr0 17796 . . . 4  |-  ( 0g
`  R )  =  ( 0g `  (oppr `  R
) )
107, 9syl6eq 2486 . . 3  |-  ( ph  ->  .0.  =  ( 0g
`  (oppr
`  R ) ) )
11 isdrngd.u . . . 4  |-  ( ph  ->  .1.  =  ( 1r
`  R ) )
12 eqid 2429 . . . . 5  |-  ( 1r
`  R )  =  ( 1r `  R
)
132, 12oppr1 17797 . . . 4  |-  ( 1r
`  R )  =  ( 1r `  (oppr `  R
) )
1411, 13syl6eq 2486 . . 3  |-  ( ph  ->  .1.  =  ( 1r
`  (oppr
`  R ) ) )
15 isdrngd.r . . . 4  |-  ( ph  ->  R  e.  Ring )
162opprring 17794 . . . 4  |-  ( R  e.  Ring  ->  (oppr `  R
)  e.  Ring )
1715, 16syl 17 . . 3  |-  ( ph  ->  (oppr
`  R )  e. 
Ring )
18 eleq1 2501 . . . . . . 7  |-  ( y  =  x  ->  (
y  e.  B  <->  x  e.  B ) )
19 neeq1 2712 . . . . . . 7  |-  ( y  =  x  ->  (
y  =/=  .0.  <->  x  =/=  .0.  ) )
2018, 19anbi12d 715 . . . . . 6  |-  ( y  =  x  ->  (
( y  e.  B  /\  y  =/=  .0.  ) 
<->  ( x  e.  B  /\  x  =/=  .0.  ) ) )
21203anbi2d 1340 . . . . 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 6312 . . . . . 6  |-  ( y  =  x  ->  (
y ( .r `  (oppr `  R ) ) z )  =  ( x ( .r `  (oppr `  R
) ) z ) )
2322neeq1d 2708 . . . . 5  |-  ( y  =  x  ->  (
( y ( .r
`  (oppr
`  R ) ) z )  =/=  .0.  <->  (
x ( .r `  (oppr `  R ) ) z )  =/=  .0.  )
)
2421, 23imbi12d 321 . . . 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 2712 . . . . . . . 8  |-  ( x  =  z  ->  (
x  =/=  .0.  <->  z  =/=  .0.  ) )
2725, 26anbi12d 715 . . . . . . 7  |-  ( x  =  z  ->  (
( x  e.  B  /\  x  =/=  .0.  ) 
<->  ( z  e.  B  /\  z  =/=  .0.  ) ) )
28273anbi3d 1341 . . . . . 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 6313 . . . . . . 7  |-  ( x  =  z  ->  (
y ( .r `  (oppr `  R ) ) x )  =  ( y ( .r `  (oppr `  R
) ) z ) )
3029neeq1d 2708 . . . . . 6  |-  ( x  =  z  ->  (
( y ( .r
`  (oppr
`  R ) ) x )  =/=  .0.  <->  (
y ( .r `  (oppr `  R ) ) z )  =/=  .0.  )
)
3128, 30imbi12d 321 . . . . 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 1026 . . . . . . . . 9  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
3433oveqd 6322 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( x  .x.  y )  =  ( x ( .r `  R ) y ) )
35 eqid 2429 . . . . . . . . 9  |-  ( .r
`  R )  =  ( .r `  R
)
36 eqid 2429 . . . . . . . . 9  |-  ( .r
`  (oppr
`  R ) )  =  ( .r `  (oppr `  R ) )
373, 35, 2, 36opprmul 17789 . . . . . . . 8  |-  ( y ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) y )
3834, 37syl6eqr 2488 . . . . . . 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 2725 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  )  /\  (
y  e.  B  /\  y  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
41403com23 1211 . . . . 5  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
x  e.  B  /\  x  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) x )  =/=  .0.  )
4231, 41chvarv 2070 . . . 4  |-  ( (
ph  /\  ( y  e.  B  /\  y  =/=  .0.  )  /\  (
z  e.  B  /\  z  =/=  .0.  ) )  ->  ( y ( .r `  (oppr `  R
) ) z )  =/=  .0.  )
4324, 42chvarv 2070 . . 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 17789 . . . 4  |-  ( I ( .r `  (oppr `  R
) ) x )  =  ( x ( .r `  R ) I )
4832adantr 466 . . . . . 6  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  ->  .x.  =  ( .r `  R ) )
4948oveqd 6322 . . . . 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 2472 . . . 4  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( x ( .r
`  R ) I )  =  .1.  )
5247, 51syl5eq 2482 . . 3  |-  ( (
ph  /\  ( x  e.  B  /\  x  =/=  .0.  ) )  -> 
( I ( .r
`  (oppr
`  R ) ) x )  =  .1.  )
535, 6, 10, 14, 17, 43, 44, 45, 46, 52isdrngd 17935 . 2  |-  ( ph  ->  (oppr
`  R )  e.  DivRing )
542opprdrng 17934 . 2  |-  ( R  e.  DivRing 
<->  (oppr
`  R )  e.  DivRing )
5553, 54sylibr 215 1  |-  ( ph  ->  R  e.  DivRing )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   ` cfv 5601  (class class class)co 6305   Basecbs 15084   .rcmulr 15153   0gc0g 15297   1rcur 17670   Ringcrg 17715  opprcoppr 17785   DivRingcdr 17910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-tpos 6981  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-0g 15299  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-grp 16624  df-minusg 16625  df-mgp 17659  df-ur 17671  df-ring 17717  df-oppr 17786  df-dvdsr 17804  df-unit 17805  df-invr 17835  df-dvr 17846  df-drng 17912
This theorem is referenced by:  erngdvlem4-rN  34275
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