rngoridm - Mathbox for Jeff Madsen

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Theorem rngoridm 39138

Description: Obsolete theorem, use ringridm 20550 instead. The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Apr-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

**Hypotheses**
Ref	Expression
uridm.1	⊢ 𝐻 = (2^nd ‘𝑅)
uridm.2	⊢ 𝑋 = ran (1^st ‘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 = (2^nd ‘𝑅)
2		uridm.2	. . 3 ⊢ 𝑋 = ran (1^st ‘𝑅)
3		uridm2.2	. . 3 ⊢ 𝑈 = (GId‘𝐻)
4	1, 2, 3	rngoidmlem 39136	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
5	4	simprd 501	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝐴𝐻𝑈) = 𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ran crn 5659 ‘cfv 6566 (class class class)co 7451 1^st c1st 8033 2^nd c2nd 8034 GIdcgi 31149 RingOpscrngo 39094

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pr 5398 ax-un 7783

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-iun 4958 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-fo 6572 df-fv 6574 df-riota 7408 df-ov 7454 df-1st 8035 df-2nd 8036 df-grpo 31152 df-gid 31153 df-ablo 31204 df-ass 39043 df-exid 39045 df-mgmOLD 39049 df-sgrOLD 39061 df-mndo 39067 df-rngo 39095

This theorem is used by: rngoueqz 39140 rngonegmn1r 39142

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