Theorem ringunitsap0 14704

Description: The set of units of a ring. If R is a local ring, # is an apartness and this theorem states that the units of a ring are those elements apart from zero (see aprlring 14710). Given the definition of #_r this theorem holds even if # is not an apartness, however. (Contributed by Jim Kingdon, 31-May-2026.)

Hypotheses

Ref	Expression
ringunitsap0.b	|- B = ( Base ` R )
ringunitsap0.z	|- .0. = ( 0g ` R )
ringunitsap0.ap	|- # = ( #r ` R )

Assertion

Ref	Expression
ringunitsap0	|- ( R e. Ring -> { x e. B | x # .0. } = (Unit ` R ) )

Distinct variable groups:   x, #   x, .0.   x, B

Allowed substitution hint:   R( x)

Proof of Theorem ringunitsap0

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4134	. . . 4 |- ( x = y -> ( x # .0. <-> y # .0. ) )
2	1	elrab 2982	. . 3 |- ( y e. { x e. B | x # .0. } <-> ( y e. B /\ y # .0. ) )
3		ringunitsap0.b	. . . 4 |- B = ( Base ` R )
4		eqid 2238	. . . 4 |- (Unit ` R ) = (Unit ` R )
5		ringunitsap0.z	. . . 4 |- .0. = ( 0g ` R )
6		ringunitsap0.ap	. . . 4 |- # = ( #r ` R )
7	3, 4, 5, 6	ringunitap 14703	. . 3 |- ( R e. Ring -> ( y e. (Unit ` R ) <-> ( y e. B /\ y # .0. ) ) )
8	2, 7	bitr4id 199	. 2 |- ( R e. Ring -> ( y e. { x e. B | x # .0. } <-> y e. (Unit ` R ) ) )
9	8	eqrdv 2236	1 |- ( R e. Ring -> { x e. B | x # .0. } = (Unit ` R ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   class class class wbr 4131   ` cfv 5380   Basecbs 13411   0gc0g 13670   Ringcrg 14408  Unitcui 14503   #_r capr 14699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-addcom 8286  ax-addass 8288  ax-i2m1 8291  ax-0lt1 8292  ax-0id 8294  ax-rnegex 8295  ax-pre-ltirr 8298  ax-pre-ltadd 8302

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8369  df-mnf 8370  df-ltxr 8372  df-inn 9315  df-2 9373  df-3 9374  df-ndx 13414  df-slot 13415  df-base 13417  df-sets 13418  df-plusg 13504  df-mulr 13505  df-0g 13672  df-mgm 13736  df-sgrp 13777  df-mnd 13790  df-grp 13868  df-minusg 13869  df-sbg 13870  df-cmn 14180  df-abl 14181  df-mgp 14326  df-ur 14371  df-srg 14376  df-ring 14410  df-dvdsr 14505  df-unit 14506  df-apr 14700

This theorem is used by: (None)