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Theorem unitrrg 17377
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 2443 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 unitrrg.u . . . . . 6  |-  U  =  (Unit `  R )
31, 2unitcl 16763 . . . . 5  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
43adantl 466 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
5 oveq2 6111 . . . . . 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 2443 . . . . . . . . . . 11  |-  ( invr `  R )  =  (
invr `  R )
7 eqid 2443 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2443 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 6, 7, 8unitlinv 16781 . . . . . . . . . 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 6118 . . . . . . . 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 16780 . . . . . . . . . 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 16673 . . . . . . . . 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 1220 . . . . . . . 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 16680 . . . . . . . . 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 2483 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( x ( .r `  R ) y ) )  =  y )
22 eqid 2443 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
231, 7, 22rngrz 16694 . . . . . . . 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 2457 . . . . . 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 2811 . . . 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 17371 . . . 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 3374 1  |-  ( R  e.  Ring  ->  U  C_  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    C_ wss 3340   ` cfv 5430  (class class class)co 6103   Basecbs 14186   .rcmulr 14251   0gc0g 14390   1rcur 16615   Ringcrg 16657  Unitcui 16743   invrcinvr 16775  RLRegcrlreg 17362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-tpos 6757  df-recs 6844  df-rdg 6878  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-nn 10335  df-2 10392  df-3 10393  df-ndx 14189  df-slot 14190  df-base 14191  df-sets 14192  df-ress 14193  df-plusg 14263  df-mulr 14264  df-0g 14392  df-mnd 15427  df-grp 15557  df-minusg 15558  df-mgp 16604  df-ur 16616  df-rng 16659  df-oppr 16727  df-dvdsr 16745  df-unit 16746  df-invr 16776  df-rlreg 17366
This theorem is referenced by:  drngdomn  17387  znrrg  18010  deg1invg  21590  ply1divalg  21621  uc1pmon1p  21635  fta1glem1  21649  ig1peu  21655  mon1psubm  29586
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