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Theorem unitrrg 13747
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 2193 . . . . . 6  |-  ( Base `  R )  =  (
Base `  R )
21a1i 9 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  ( Base `  R )  =  ( Base `  R
) )
3 unitrrg.u . . . . . 6  |-  U  =  (Unit `  R )
43a1i 9 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  U  =  (Unit `  R )
)
5 ringsrg 13527 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. SRing
)
65adantr 276 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  R  e. SRing )
7 simpr 110 . . . . 5  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  U )
82, 4, 6, 7unitcld 13588 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  ( Base `  R
) )
9 oveq2 5918 . . . . . 6  |-  ( ( x ( .r `  R ) y )  =  ( 0g `  R )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) ( x ( .r `  R ) y ) )  =  ( ( ( invr `  R ) `  x
) ( .r `  R ) ( 0g
`  R ) ) )
10 eqid 2193 . . . . . . . . . . 11  |-  ( invr `  R )  =  (
invr `  R )
11 eqid 2193 . . . . . . . . . . 11  |-  ( .r
`  R )  =  ( .r `  R
)
12 eqid 2193 . . . . . . . . . . 11  |-  ( 1r
`  R )  =  ( 1r `  R
)
133, 10, 11, 12unitlinv 13606 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( ( invr `  R
) `  x )
( .r `  R
) x )  =  ( 1r `  R
) )
1413adantr 276 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) x )  =  ( 1r
`  R ) )
1514oveq1d 5925 . . . . . . . 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 ) )
16 simpll 527 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  R  e.  Ring )
173, 10, 1ringinvcl 13605 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  (
( invr `  R ) `  x )  e.  (
Base `  R )
)
1817adantr 276 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( invr `  R ) `  x )  e.  (
Base `  R )
)
198adantr 276 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  x  e.  ( Base `  R )
)
20 simpr 110 . . . . . . . . 9  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  y  e.  ( Base `  R )
)
211, 11ringass 13496 . . . . . . . . 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 ) ) )
2216, 18, 19, 20, 21syl13anc 1251 . . . . . . . 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 ) ) )
231, 11, 12ringlidm 13503 . . . . . . . . 9  |-  ( ( R  e.  Ring  /\  y  e.  ( Base `  R
) )  ->  (
( 1r `  R
) ( .r `  R ) y )  =  y )
2423adantlr 477 . . . . . . . 8  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( ( 1r `  R ) ( .r `  R ) y )  =  y )
2515, 22, 243eqtr3d 2234 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( x ( .r `  R ) y ) )  =  y )
26 eqid 2193 . . . . . . . . 9  |-  ( 0g
`  R )  =  ( 0g `  R
)
271, 11, 26ringrz 13524 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  (
( invr `  R ) `  x )  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2816, 18, 27syl2anc 411 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
( invr `  R ) `  x ) ( .r
`  R ) ( 0g `  R ) )  =  ( 0g
`  R ) )
2925, 28eqeq12d 2208 . . . . . 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 ) ) )
309, 29imbitrid 154 . . . . 5  |-  ( ( ( R  e.  Ring  /\  x  e.  U )  /\  y  e.  (
Base `  R )
)  ->  ( (
x ( .r `  R ) y )  =  ( 0g `  R )  ->  y  =  ( 0g `  R ) ) )
3130ralrimiva 2567 . . . 4  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  ( 0g `  R
)  ->  y  =  ( 0g `  R ) ) )
32 unitrrg.e . . . . 5  |-  E  =  (RLReg `  R )
3332, 1, 11, 26isrrg 13743 . . . 4  |-  ( x  e.  E  <->  ( x  e.  ( Base `  R
)  /\  A. y  e.  ( Base `  R
) ( ( x ( .r `  R
) y )  =  ( 0g `  R
)  ->  y  =  ( 0g `  R ) ) ) )
348, 31, 33sylanbrc 417 . . 3  |-  ( ( R  e.  Ring  /\  x  e.  U )  ->  x  e.  E )
3534ex 115 . 2  |-  ( R  e.  Ring  ->  ( x  e.  U  ->  x  e.  E ) )
3635ssrdv 3185 1  |-  ( R  e.  Ring  ->  U  C_  E )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2164   A.wral 2472    C_ wss 3153   ` cfv 5246  (class class class)co 5910   Basecbs 12608   .rcmulr 12686   0gc0g 12857   1rcur 13439  SRingcsrg 13443   Ringcrg 13476  Unitcui 13567   invrcinvr 13600  RLRegcrlreg 13735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-setind 4565  ax-cnex 7953  ax-resscn 7954  ax-1cn 7955  ax-1re 7956  ax-icn 7957  ax-addcl 7958  ax-addrcl 7959  ax-mulcl 7960  ax-addcom 7962  ax-addass 7964  ax-i2m1 7967  ax-0lt1 7968  ax-0id 7970  ax-rnegex 7971  ax-pre-ltirr 7974  ax-pre-lttrn 7976  ax-pre-ltadd 7978
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-f1 5251  df-fo 5252  df-f1o 5253  df-fv 5254  df-riota 5865  df-ov 5913  df-oprab 5914  df-mpo 5915  df-tpos 6289  df-pnf 8046  df-mnf 8047  df-ltxr 8049  df-inn 8973  df-2 9031  df-3 9032  df-ndx 12611  df-slot 12612  df-base 12614  df-sets 12615  df-iress 12616  df-plusg 12698  df-mulr 12699  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988  df-grp 13065  df-minusg 13066  df-cmn 13345  df-abl 13346  df-mgp 13401  df-ur 13440  df-srg 13444  df-ring 13478  df-oppr 13548  df-dvdsr 13569  df-unit 13570  df-invr 13601  df-rlreg 13738
This theorem is referenced by:  znrrg  14125
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