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Theorem unitrrg 18137
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 2454 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
2 unitrrg.u . . . . . 6  |-  U  =  (Unit `  R )
31, 2unitcl 17503 . . . . 5  |-  ( x  e.  U  ->  x  e.  ( Base `  R
) )
43adantl 464 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
5 oveq2 6278 . . . . . 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 2454 . . . . . . . . . . 11  |-  ( invr `  R )  =  (
invr `  R )
7 eqid 2454 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
8 eqid 2454 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
92, 6, 7, 8unitlinv 17521 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
109adantr 463 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) x )  =  ( 1r
`  R ) )
1110oveq1d 6285 . . . . . . . 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 751 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  R  e.  Ring )
132, 6, 1ringinvcl 17520 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( invr `  R ) `  x )  e.  (
Base `  R )
)
1413adantr 463 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( invr `  R ) `  x )  e.  (
Base `  R )
)
154adantr 463 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  x  e.  ( Base `  R )
)
16 simpr 459 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  y  e.  ( Base `  R )
)
171, 7ringass 17410 . . . . . . . . 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 1228 . . . . . . . 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, 8ringlidm 17417 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) y )  =  y )
2019adantlr 712 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( 1r `  R ) ( .r `  R ) y )  =  y )
2111, 18, 203eqtr3d 2503 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( x ( .r `  R ) y ) )  =  y )
22 eqid 2454 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
231, 7, 22ringrz 17431 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  x )  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2412, 14, 23syl2anc 659 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2521, 24eqeq12d 2476 . . . . . 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 2868 . . . 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 18131 . . . 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 662 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  E )
3130ex 432 . 2  |-  ( R  e.  Ring  ->  ( x  e.  U  ->  x  e.  E ) )
3231ssrdv 3495 1  |-  ( R  e.  Ring  ->  U  C_  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804    C_ wss 3461   ` cfv 5570  (class class class)co 6270   Basecbs 14716   .rcmulr 14785   0gc0g 14929   1rcur 17348   Ringcrg 17393  Unitcui 17483   invrcinvr 17515  RLRegcrlreg 18122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-ress 14723  df-plusg 14797  df-mulr 14798  df-0g 14931  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-mgp 17337  df-ur 17349  df-ring 17395  df-oppr 17467  df-dvdsr 17485  df-unit 17486  df-invr 17516  df-rlreg 18126
This theorem is referenced by:  drngdomn  18147  znrrg  18777  deg1invg  22673  ply1divalg  22704  uc1pmon1p  22718  fta1glem1  22732  ig1peu  22738  mon1psubm  31407
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