Theorem ringunitsap0 14454

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 14460). 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 4114	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 # 0 ↔ 𝑦 # 0 ))
2	1	elrab 2975	. . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝑥 # 0 } ↔ (𝑦 ∈ 𝐵 ∧ 𝑦 # 0 ))
3		ringunitsap0.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
4		eqid 2234	. . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅)
5		ringunitsap0.z	. . . 4 ⊢ 0 = (0g‘𝑅)
6		ringunitsap0.ap	. . . 4 ⊢ # = (#r‘𝑅)
7	3, 4, 5, 6	ringunitap 14453	. . 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 2232	1 ⊢ (𝑅 ∈ Ring → {𝑥 ∈ 𝐵 ∣ 𝑥 # 0 } = (Unit‘𝑅))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2205  {crab 2526   class class class wbr 4111  ‘cfv 5354  Basecbs 13233  0_gc0g 13490  Ringcrg 14161  Unitcui 14253  #_rcapr 14449

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-addcom 8232  ax-addass 8234  ax-i2m1 8237  ax-0lt1 8238  ax-0id 8240  ax-rnegex 8241  ax-pre-ltirr 8244  ax-pre-ltadd 8248

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8315  df-mnf 8316  df-ltxr 8318  df-inn 9243  df-2 9301  df-3 9302  df-ndx 13236  df-slot 13237  df-base 13239  df-sets 13240  df-plusg 13324  df-mulr 13325  df-0g 13492  df-mgm 13590  df-sgrp 13636  df-mnd 13651  df-grp 13737  df-minusg 13738  df-sbg 13739  df-cmn 14024  df-abl 14025  df-mgp 14086  df-ur 14125  df-srg 14129  df-ring 14163  df-dvdsr 14255  df-unit 14256  df-apr 14450

This theorem is referenced by: (None)