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Theorem unitrrg 17753
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 2467 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 unitrrg.u . . . . . 6  |-  U  =  (Unit `  R )
31, 2unitcl 17121 . . . . 5  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
43adantl 466 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
5 oveq2 6293 . . . . . 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 2467 . . . . . . . . . . 11  |-  ( invr `  R )  =  (
invr `  R )
7 eqid 2467 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2467 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 6, 7, 8unitlinv 17139 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
109adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) x )  =  ( 1r
`  R ) )
1110oveq1d 6300 . . . . . . . 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 753 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  R  e.  Ring )
132, 6, 1rnginvcl 17138 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( invr `  R ) `  x )  e.  (
Base `  R )
)
1413adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( invr `  R ) `  x )  e.  (
Base `  R )
)
154adantr 465 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  x  e.  ( Base `  R )
)
16 simpr 461 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  y  e.  ( Base `  R )
)
171, 7rngass 17028 . . . . . . . . 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 1230 . . . . . . . 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 17035 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) y )  =  y )
2019adantlr 714 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( 1r `  R ) ( .r `  R ) y )  =  y )
2111, 18, 203eqtr3d 2516 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( x ( .r `  R ) y ) )  =  y )
22 eqid 2467 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
231, 7, 22rngrz 17049 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  x )  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2412, 14, 23syl2anc 661 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2521, 24eqeq12d 2489 . . . . . 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 219 . . . . 5  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) )
2726ralrimiva 2878 . . . 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 17747 . . . 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 664 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  E )
3130ex 434 . 2  |-  ( R  e.  Ring  ->  ( x  e.  U  ->  x  e.  E ) )
3231ssrdv 3510 1  |-  ( R  e.  Ring  ->  U  C_  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   ` cfv 5588  (class class class)co 6285   Basecbs 14493   .rcmulr 14559   0gc0g 14698   1rcur 16967   Ringcrg 17012  Unitcui 17101   invrcinvr 17133  RLRegcrlreg 17738
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-mulcom 9557  ax-addass 9558  ax-mulass 9559  ax-distr 9560  ax-i2m1 9561  ax-1ne0 9562  ax-1rid 9563  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568  ax-pre-ltadd 9569  ax-pre-mulgt0 9570
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 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-om 6686  df-tpos 6956  df-recs 7043  df-rdg 7077  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-sub 9808  df-neg 9809  df-nn 10538  df-2 10595  df-3 10596  df-ndx 14496  df-slot 14497  df-base 14498  df-sets 14499  df-ress 14500  df-plusg 14571  df-mulr 14572  df-0g 14700  df-mnd 15735  df-grp 15871  df-minusg 15872  df-mgp 16956  df-ur 16968  df-rng 17014  df-oppr 17085  df-dvdsr 17103  df-unit 17104  df-invr 17134  df-rlreg 17742
This theorem is referenced by:  drngdomn  17763  znrrg  18411  deg1invg  22334  ply1divalg  22365  uc1pmon1p  22379  fta1glem1  22393  ig1peu  22399  mon1psubm  30998
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