rngoidmlem - Mathbox for Jeff Madsen

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Theorem rngoidmlem 38850

Description: Obsolete theorem, use ringidmlem 20490 instead. The unity element of a ring is an identity element for the multiplication. (Contributed by FL, 18-Feb-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)
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 38849	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3		mndomgmid 38785	. . . 4 ⊢ (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
4		eqid 2761	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
5		uridm.3	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
6	4, 5	cmpidelt 38773	. . . . 5 ⊢ ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
7	6	ex 418	. . . 4 ⊢ (𝐻 ∈ (Magma ∩ ExId ) → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
8	2, 3, 7	3syl 19	. . 3 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
9		eqid 2761	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
10	1, 9	rngorn1eq 38848	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = ran 𝐻)
11		uridm.2	. . . . 5 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		eqtr 2781	. . . . . 6 ⊢ ((𝑋 = ran (1^st ‘𝑅) ∧ ran (1^st ‘𝑅) = ran 𝐻) → 𝑋 = ran 𝐻)
13		simpl 488	. . . . . . . . 9 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → 𝑋 = ran 𝐻)
14	13	eleq2d 2847	. . . . . . . 8 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ran 𝐻))
15	14	imbi1d 344	. . . . . . 7 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
16	15	ex 418	. . . . . 6 ⊢ (𝑋 = ran 𝐻 → (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))))
17	12, 16	syl 18	. . . . 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 703	. . . 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 39	. . 3 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
20	8, 19	mpbird 260	. 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
21	20	imp 412	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∩ cin 3898 ran crn 5652 ‘cfv 6537 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 GIdcgi 31085 ExId cexid 38758 Magmacmagm 38762 MndOpcmndo 38780 RingOpscrngo 38808

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pr 5391 ax-un 7749

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-fo 6543 df-fv 6545 df-riota 7375 df-ov 7421 df-1st 7999 df-2nd 8000 df-grpo 31088 df-gid 31089 df-ablo 31140 df-ass 38757 df-exid 38759 df-mgmOLD 38763 df-sgrOLD 38775 df-mndo 38781 df-rngo 38809

This theorem is used by: rngolidm 38851 rngoridm 38852

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