rngolidm - Mathbox for Jeff Madsen

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Theorem rngolidm 34222

Description: The unit 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 34221	. 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
5	4	simpld 489	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (𝑈𝐻𝐴) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ran crn 5314 ‘cfv 6102 (class class class)co 6879 1^st c1st 7400 2^nd c2nd 7401 GIdcgi 27869 RingOpscrngo 34179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-nul 4117 df-if 4279 df-sn 4370 df-pr 4372 df-op 4376 df-uni 4630 df-iun 4713 df-br 4845 df-opab 4907 df-mpt 4924 df-id 5221 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-fo 6108 df-fv 6110 df-riota 6840 df-ov 6882 df-1st 7402 df-2nd 7403 df-grpo 27872 df-gid 27873 df-ablo 27924 df-ass 34128 df-exid 34130 df-mgmOLD 34134 df-sgrOLD 34146 df-mndo 34152 df-rngo 34180

This theorem is referenced by: rngonegmn1l 34226 zerdivemp1x 34232 isdrngo2 34243 1idl 34311 smprngopr 34337 prnc 34352

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