rngoidmlem - Mathbox for Jeff Madsen

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Theorem rngoidmlem 32905

Description: The unit of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-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
rngoidmlem	⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

**Proof of Theorem rngoidmlem**
Step	Hyp	Ref	Expression
1		uridm.1	. . . . 5 ⊢ 𝐻 = (2^nd ‘𝑅)
2	1	rngomndo 32904	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3		mndomgmid 32840	. . . 4 ⊢ (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
4		eqid 2610	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
5		uridm.3	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
6	4, 5	cmpidelt 32828	. . . . 5 ⊢ ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
7	6	ex 449	. . . 4 ⊢ (𝐻 ∈ (Magma ∩ ExId ) → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
8	2, 3, 7	3syl 18	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
9		eqid 2610	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
10	1, 9	rngorn1eq 32903	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = ran 𝐻)
11		uridm.2	. . . . 5 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		eqtr 2629	. . . . . 6 ⊢ ((𝑋 = ran (1^st ‘𝑅) ∧ ran (1^st ‘𝑅) = ran 𝐻) → 𝑋 = ran 𝐻)
13		simpl 472	. . . . . . . . 9 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → 𝑋 = ran 𝐻)
14	13	eleq2d 2673	. . . . . . . 8 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ran 𝐻))
15	14	imbi1d 330	. . . . . . 7 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
16	15	ex 449	. . . . . 6 ⊢ (𝑋 = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
17	12, 16	syl 17	. . . . 5 ⊢ ((𝑋 = ran (1^st ‘𝑅) ∧ ran (1^st ‘𝑅) = ran 𝐻) → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
18	11, 17	mpan 702	. . . 4 ⊢ (ran (1^st ‘𝑅) = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
19	10, 18	mpcom 37	. . 3 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
20	8, 19	mpbird 246	. 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
21	20	imp 444	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∩ cin 3539 ran crn 5039 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 GIdcgi 26728 ExId cexid 32813 Magmacmagm 32817 MndOpcmndo 32835 RingOpscrngo 32863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-riota 6511 df-ov 6552 df-1st 7059 df-2nd 7060 df-grpo 26731 df-gid 26732 df-ablo 26783 df-ass 32812 df-exid 32814 df-mgmOLD 32818 df-sgrOLD 32830 df-mndo 32836 df-rngo 32864

This theorem is referenced by: rngolidm 32906 rngoridm 32907

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