Theorem rngoridm 38135

Description: The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
uridm.1	⊢ 𝐻 = (2nd ‘𝑅)
uridm.2	⊢ 𝑋 = ran (1st ‘𝑅)
uridm2.2	⊢ 𝑈 = (GId‘𝐻)

Assertion

Ref	Expression
rngoridm	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)

Proof of Theorem rngoridm

Step	Hyp	Ref	Expression
1		uridm.1	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
2		uridm.2	. . 3 ⊢ 𝑋 = ran (1st ‘𝑅)
3		uridm2.2	. . 3 ⊢ 𝑈 = (GId‘𝐻)
4	1, 2, 3	rngoidmlem 38133	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
5	4	simprd 495	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ran crn 5625  ‘cfv 6492  (class class class)co 7358  1^st c1st 7931  2^nd c2nd 7932  GIdcgi 30565  RingOpscrngo 38091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-riota 7315  df-ov 7361  df-1st 7933  df-2nd 7934  df-grpo 30568  df-gid 30569  df-ablo 30620  df-ass 38040  df-exid 38042  df-mgmOLD 38046  df-sgrOLD 38058  df-mndo 38064  df-rngo 38092

This theorem is referenced by:  rngoueqz  38137  rngonegmn1r  38139