rngolidm - Mathbox for Jeff Madsen

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Theorem rngolidm 37987

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 = (2^nd ‘𝑅)
uridm.2	⊢ 𝑋 = ran (1^st ‘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 = (2^nd ‘𝑅)
2		uridm.2	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
3		uridm.3	. . 3 ⊢ 𝑈 = (GId‘𝐻)
4	1, 2, 3	rngoidmlem 37986	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
5	4	simpld 494	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ran crn 5615 ‘cfv 6481 (class class class)co 7346 1^st c1st 7919 2^nd c2nd 7920 GIdcgi 30470 RingOpscrngo 37944

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4281 df-if 4473 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-fo 6487 df-fv 6489 df-riota 7303 df-ov 7349 df-1st 7921 df-2nd 7922 df-grpo 30473 df-gid 30474 df-ablo 30525 df-ass 37893 df-exid 37895 df-mgmOLD 37899 df-sgrOLD 37911 df-mndo 37917 df-rngo 37945

This theorem is referenced by: rngonegmn1l 37991 zerdivemp1x 37997 isdrngo2 38008 1idl 38076 smprngopr 38102 prnc 38117

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