rngoidmlem - Mathbox for Jeff Madsen

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Theorem rngoidmlem 38396

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 38395	. . . 4 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)
3		mndomgmid 38331	. . . 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 38319	. . . . 5 ⊢ ((𝐻 ∈ (Magma ∩ ExId ) ∧ 𝐴 ∈ ran 𝐻) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))
7	6	ex 416	. . . 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 2761	. . . . 5 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
10	1, 9	rngorn1eq 38394	. . . 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 486	. . . . . . . . 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 343	. . . . . . 7 ⊢ ((𝑋 = ran 𝐻 ∧ 𝑅 ∈ RingOps) → ((𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)) ↔ (𝐴 ∈ ran 𝐻 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))))
16	15	ex 416	. . . . . 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 700	. . . 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 259	. 2 ⊢ (𝑅 ∈ RingOps → (𝐴 ∈ 𝑋 → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴)))
21	20	imp 410	1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ((𝑈𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑈) = 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∩ cin 3901 ran crn 5644 ‘cfv 6516 (class class class)co 7391 1^st c1st 7963 2^nd c2nd 7964 GIdcgi 30650 ExId cexid 38304 Magmacmagm 38308 MndOpcmndo 38326 RingOpscrngo 38354

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5243 ax-nul 5253 ax-pr 5387 ax-un 7713

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4284 df-if 4478 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-id 5538 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-fo 6522 df-fv 6524 df-riota 7348 df-ov 7394 df-1st 7965 df-2nd 7966 df-grpo 30653 df-gid 30654 df-ablo 30705 df-ass 38303 df-exid 38305 df-mgmOLD 38309 df-sgrOLD 38321 df-mndo 38327 df-rngo 38355

This theorem is referenced by: rngolidm 38397 rngoridm 38398

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