rngolidm - Mathbox for Jeff Madsen

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Theorem rngolidm 37944

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 37943	. 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 1540 ∈ wcel 2108 ran crn 5686 ‘cfv 6561 (class class class)co 7431 1^st c1st 8012 2^nd c2nd 8013 GIdcgi 30509 RingOpscrngo 37901

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fo 6567 df-fv 6569 df-riota 7388 df-ov 7434 df-1st 8014 df-2nd 8015 df-grpo 30512 df-gid 30513 df-ablo 30564 df-ass 37850 df-exid 37852 df-mgmOLD 37856 df-sgrOLD 37868 df-mndo 37874 df-rngo 37902

This theorem is referenced by: rngonegmn1l 37948 zerdivemp1x 37954 isdrngo2 37965 1idl 38033 smprngopr 38059 prnc 38074

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