Theorem ringunitsap0 14703

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 14709). 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 ∈ Ring → {𝑥 ∈ 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 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 # 0 } ↔ (𝑦 ∈ 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 14702	. . 3 ⊢ (𝑅 ∈ Ring → (𝑦 ∈ (Unit‘𝑅) ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 # 0 )))
8	2, 7	bitr4id 199	. 2 ⊢ (𝑅 ∈ Ring → (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 # 0 } ↔ 𝑦 ∈ (Unit‘𝑅)))
9	8	eqrdv 2236	1 ⊢ (𝑅 ∈ Ring → {𝑥 ∈ 𝐵 ∣ 𝑥 # 0 } = (Unit‘𝑅))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {crab 2532   class class class wbr 4131  ‘cfv 5380  Basecbs 13410  0_gc0g 13669  Ringcrg 14407  Unitcui 14502   #_r capr 14698

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 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-pre-ltirr 8297  ax-pre-ltadd 8301

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 8368  df-mnf 8369  df-ltxr 8371  df-inn 9314  df-2 9372  df-3 9373  df-ndx 13413  df-slot 13414  df-base 13416  df-sets 13417  df-plusg 13503  df-mulr 13504  df-0g 13671  df-mgm 13735  df-sgrp 13776  df-mnd 13789  df-grp 13867  df-minusg 13868  df-sbg 13869  df-cmn 14179  df-abl 14180  df-mgp 14325  df-ur 14370  df-srg 14375  df-ring 14409  df-dvdsr 14504  df-unit 14505  df-apr 14699

This theorem is used by: (None)