Theorem rngolidm 38648

Description: Obsolete theorem, use ringgrp 20366 instead. The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
uridm.1	⊢ 𝐻 = (2nd ‘𝑅)
uridm.2	⊢ 𝑋 = ran (1st ‘𝑅)
uridm.3	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
rngolidm	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)

Proof of Theorem rngolidm

Step	Hyp	Ref	Expression
1		uridm.1	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
2		uridm.2	. . 3 ⊢ 𝑋 = ran (1st ‘𝑅)
3		uridm.3	. . 3 ⊢ 𝑈 = (GId‘𝐻)
4	1, 2, 3	rngoidmlem 38647	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
5	4	simpld 500	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ran crn 5664  ‘cfv 6540  (class class class)co 7419  1^st c1st 7990  2^nd c2nd 7991  GIdcgi 30915  RingOpscrngo 38605

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-riota 7376  df-ov 7422  df-1st 7992  df-2nd 7993  df-grpo 30918  df-gid 30919  df-ablo 30970  df-ass 38554  df-exid 38556  df-mgmOLD 38560  df-sgrOLD 38572  df-mndo 38578  df-rngo 38606

This theorem is used by:  rngonegmn1l  38652  zerdivemp1x  38658  isdrngo2  38669  1idl  38737  smprngopr  38763  prnc  38778