rngoidmlem - Mathbox for Jeff Madsen

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Theorem rngoidmlem 38137

Description: The unity element 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 38136	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3		mndomgmid 38072	. . . 4 ⊢ (𝐻 ∈ MndOp → 𝐻 ∈ (Magma ∩ ExId ))
4		eqid 2736	. . . . . 6 ⊢ ran 𝐻 = ran 𝐻
5		uridm.3	. . . . . 6 ⊢ 𝑈 = (GId‘𝐻)
6	4, 5	cmpidelt 38060	. . . . 5 ⊢ ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
7	6	ex 412	. . . 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 2736	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
10	1, 9	rngorn1eq 38135	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = ran 𝐻)
11		uridm.2	. . . . 5 ⊢ 𝑋 = ran (1^st ‘𝑅)
12		eqtr 2756	. . . . . 6 ⊢ ((𝑋 = ran (1^st ‘𝑅) ∧ ran (1^st ‘𝑅) = ran 𝐻) → 𝑋 = ran 𝐻)
13		simpl 482	. . . . . . . . 9 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → 𝑋 = ran 𝐻)
14	13	eleq2d 2822	. . . . . . . 8 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → (𝐴 ∈ 𝑋 ↔ 𝐴 ∈ ran 𝐻))
15	14	imbi1d 341	. . . . . . 7 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
16	15	ex 412	. . . . . 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 690	. . . 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 38	. . 3 ⊢ (𝑅 ∈ RingOps → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
20	8, 19	mpbird 257	. 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
21	20	imp 406	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∩ cin 3900 ran crn 5625 ‘cfv 6492 (class class class)co 7358 1^st c1st 7931 2^nd c2nd 7932 GIdcgi 30565 ExId cexid 38045 Magmacmagm 38049 MndOpcmndo 38067 RingOpscrngo 38095

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-fo 6498 df-fv 6500 df-riota 7315 df-ov 7361 df-1st 7933 df-2nd 7934 df-grpo 30568 df-gid 30569 df-ablo 30620 df-ass 38044 df-exid 38046 df-mgmOLD 38050 df-sgrOLD 38062 df-mndo 38068 df-rngo 38096

This theorem is referenced by: rngolidm 38138 rngoridm 38139

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