rngolidm - Mathbox for Jeff Madsen

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Theorem rngolidm 36800

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 36799	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
5	4	simpld 495	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ran crn 5677 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 GIdcgi 29738 RingOpscrngo 36757

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-riota 7364 df-ov 7411 df-1st 7974 df-2nd 7975 df-grpo 29741 df-gid 29742 df-ablo 29793 df-ass 36706 df-exid 36708 df-mgmOLD 36712 df-sgrOLD 36724 df-mndo 36730 df-rngo 36758

This theorem is referenced by: rngonegmn1l 36804 zerdivemp1x 36810 isdrngo2 36821 1idl 36889 smprngopr 36915 prnc 36930

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