Theorem rngoridm 38259

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 38257	. 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 1542   ∈ wcel 2114  ran crn 5632  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  GIdcgi 30561  RingOpscrngo 38215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  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 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-fo 6504  df-fv 6506  df-riota 7324  df-ov 7370  df-1st 7942  df-2nd 7943  df-grpo 30564  df-gid 30565  df-ablo 30616  df-ass 38164  df-exid 38166  df-mgmOLD 38170  df-sgrOLD 38182  df-mndo 38188  df-rngo 38216

This theorem is referenced by:  rngoueqz  38261  rngonegmn1r  38263