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Theorem unitrrg 16050
Description: Units are regular elements. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Hypotheses
Ref Expression
unitrrg.e  |-  E  =  (RLReg `  R )
unitrrg.u  |-  U  =  (Unit `  R )
Assertion
Ref Expression
unitrrg  |-  ( R  e.  Ring  ->  U  C_  E )

Proof of Theorem unitrrg
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 unitrrg.u . . . . . 6  |-  U  =  (Unit `  R )
31, 2unitcl 15457 . . . . 5  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
43adantl 452 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
5 oveq2 5882 . . . . . 6  |-  ( ( x ( .r `  R ) y )  =  ( 0g `  R )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) ( x ( .r `  R ) y ) )  =  ( ( ( invr `  R ) `  x
) ( .r `  R ) ( 0g
`  R ) ) )
6 eqid 2296 . . . . . . . . . . 11  |-  ( invr `  R )  =  (
invr `  R )
7 eqid 2296 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2296 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 6, 7, 8unitlinv 15475 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
109adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) x )  =  ( 1r
`  R ) )
1110oveq1d 5889 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( ( invr `  R
) `  x )
( .r `  R
) x ) ( .r `  R ) y )  =  ( ( 1r `  R
) ( .r `  R ) y ) )
12 simpll 730 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  R  e.  Ring )
132, 6, 1rnginvcl 15474 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( invr `  R ) `  x )  e.  (
Base `  R )
)
1413adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( invr `  R ) `  x )  e.  (
Base `  R )
)
154adantr 451 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  x  e.  ( Base `  R )
)
16 simpr 447 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  y  e.  ( Base `  R )
)
171, 7rngass 15373 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  (
( ( invr `  R
) `  x )  e.  ( Base `  R
)  /\  x  e.  ( Base `  R )  /\  y  e.  ( Base `  R ) ) )  ->  ( (
( ( invr `  R
) `  x )
( .r `  R
) x ) ( .r `  R ) y )  =  ( ( ( invr `  R
) `  x )
( .r `  R
) ( x ( .r `  R ) y ) ) )
1812, 14, 15, 16, 17syl13anc 1184 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( ( invr `  R
) `  x )
( .r `  R
) x ) ( .r `  R ) y )  =  ( ( ( invr `  R
) `  x )
( .r `  R
) ( x ( .r `  R ) y ) ) )
191, 7, 8rnglidm 15380 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) y )  =  y )
2019adantlr 695 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( 1r `  R ) ( .r `  R ) y )  =  y )
2111, 18, 203eqtr3d 2336 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( x ( .r `  R ) y ) )  =  y )
22 eqid 2296 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
231, 7, 22rngrz 15394 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  x )  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2412, 14, 23syl2anc 642 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2521, 24eqeq12d 2310 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( ( invr `  R
) `  x )
( .r `  R
) ( x ( .r `  R ) y ) )  =  ( ( ( invr `  R ) `  x
) ( .r `  R ) ( 0g
`  R ) )  <-> 
y  =  ( 0g
`  R ) ) )
265, 25syl5ib 210 . . . . 5  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) )
2726ralrimiva 2639 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  ( 0g `  R
)  ->  y  =  ( 0g `  R ) ) )
28 unitrrg.e . . . . 5  |-  E  =  (RLReg `  R )
2928, 1, 7, 22isrrg 16045 . . . 4  |-  ( x  e.  E  <->  ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  ( 0g `  R
)  ->  y  =  ( 0g `  R ) ) ) )
304, 27, 29sylanbrc 645 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  E )
3130ex 423 . 2  |-  ( R  e.  Ring  ->  ( x  e.  U  ->  x  e.  E ) )
3231ssrdv 3198 1  |-  ( R  e.  Ring  ->  U  C_  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   .rcmulr 13225   0gc0g 13416   Ringcrg 15353   1rcur 15355  Unitcui 15437   invrcinvr 15469  RLRegcrlreg 16036
This theorem is referenced by:  drngdomn  16060  znrrg  16535  deg1invg  19508  ply1divalg  19539  uc1pmon1p  19553  fta1glem1  19567  ig1peu  19573  mon1psubm  27628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-mulr 13238  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-mgp 15342  df-rng 15356  df-ur 15358  df-oppr 15421  df-dvdsr 15439  df-unit 15440  df-invr 15470  df-rlreg 16040
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