rngomndo - Mathbox for Jeff Madsen

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Theorem rngomndo 36020

Description: In a unital ring the multiplication is a monoid. (Contributed by FL, 24-Jan-2010.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
unmnd.1	⊢ 𝐻 = (2^nd ‘𝑅)

**Assertion**
Ref	Expression
rngomndo	⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Proof of Theorem rngomndo Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		unmnd.1	. . . 4 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2738	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4	1, 2, 3	rngosm 35985	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅))
5	1, 2, 3	rngoass 35991	. . . 4 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ ran (1^st ‘𝑅) ∧ 𝑦 ∈ ran (1^st ‘𝑅) ∧ 𝑧 ∈ ran (1^st ‘𝑅))) → ((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
6	5	ralrimivvva 3115	. . 3 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)))
7	1, 2, 3	rngoi 35984	. . . 4 ⊢ (𝑅 ∈ RingOps → (((1^st ‘𝑅) ∈ AbelOp ∧ 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅)) ∧ (∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦(1^st ‘𝑅)𝑧)) = ((𝑥𝐻𝑦)(1^st ‘𝑅)(𝑥𝐻𝑧)) ∧ ((𝑥(1^st ‘𝑅)𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)(1^st ‘𝑅)(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
8	7	simprrd 770	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
9	2, 1	rngorn1 36018	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = dom dom 𝐻)
10		xpid11 5830	. . . . . . . 8 ⊢ ((dom dom 𝐻 × dom dom 𝐻) = (ran (1^st ‘𝑅) × ran (1^st ‘𝑅)) ↔ dom dom 𝐻 = ran (1^st ‘𝑅))
11	10	biimpri 227	. . . . . . 7 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1^st ‘𝑅) × ran (1^st ‘𝑅)))
12		feq23 6568	. . . . . . 7 ⊢ (((dom dom 𝐻 × dom dom 𝐻) = (ran (1^st ‘𝑅) × ran (1^st ‘𝑅)) ∧ dom dom 𝐻 = ran (1^st ‘𝑅)) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ↔ 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅)))
13	11, 12	mpancom 684	. . . . . 6 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ↔ 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅)))
14		raleq 3333	. . . . . . . 8 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
15	14	raleqbi1dv 3331	. . . . . . 7 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
16	15	raleqbi1dv 3331	. . . . . 6 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ↔ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧))))
17		raleq 3333	. . . . . . 7 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
18	17	rexeqbi1dv 3332	. . . . . 6 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦) ↔ ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
19	13, 16, 18	3anbi123d 1434	. . . . 5 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → ((𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) ↔ (𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅) ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
20	19	eqcoms 2746	. . . 4 ⊢ (ran (1^st ‘𝑅) = dom dom 𝐻 → ((𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) ↔ (𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅) ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
21	9, 20	syl 17	. . 3 ⊢ (𝑅 ∈ RingOps → ((𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) ↔ (𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅) ∧ ∀𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)∀𝑧 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
22	4, 6, 8, 21	mpbir3and 1340	. 2 ⊢ (𝑅 ∈ RingOps → (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
23		fvex 6769	. . . 4 ⊢ (2^nd ‘𝑅) ∈ V
24		eleq1 2826	. . . 4 ⊢ (𝐻 = (2^nd ‘𝑅) → (𝐻 ∈ V ↔ (2^nd ‘𝑅) ∈ V))
25	23, 24	mpbiri 257	. . 3 ⊢ (𝐻 = (2^nd ‘𝑅) → 𝐻 ∈ V)
26		eqid 2738	. . . 4 ⊢ dom dom 𝐻 = dom dom 𝐻
27	26	ismndo1 35958	. . 3 ⊢ (𝐻 ∈ V → (𝐻 ∈ MndOp ↔ (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))))
28	2, 25, 27	mp2b 10	. 2 ⊢ (𝐻 ∈ MndOp ↔ (𝐻:(dom dom 𝐻 × dom dom 𝐻)⟶dom dom 𝐻 ∧ ∀𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻∀𝑧 ∈ dom dom 𝐻((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ ∃𝑥 ∈ dom dom 𝐻∀𝑦 ∈ dom dom 𝐻((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))
29	22, 28	sylibr 233	1 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 × cxp 5578 dom cdm 5580 ran crn 5581 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 1^st c1st 7802 2^nd c2nd 7803 AbelOpcablo 28807 MndOpcmndo 35951 RingOpscrngo 35979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pr 5347 ax-un 7566

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-fo 6424 df-fv 6426 df-ov 7258 df-1st 7804 df-2nd 7805 df-grpo 28756 df-ablo 28808 df-ass 35928 df-exid 35930 df-mgmOLD 35934 df-sgrOLD 35946 df-mndo 35952 df-rngo 35980

This theorem is referenced by: rngoidmlem 36021 rngo1cl 36024 isdrngo2 36043

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