rngomndo - Mathbox for Jeff Madsen

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Theorem rngomndo 37929

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 2729	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		unmnd.1	. . . 4 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2729	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4	1, 2, 3	rngosm 37894	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅))
5	1, 2, 3	rngoass 37900	. . . 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 3183	. . 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 37893	. . . 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 773	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
9	2, 1	rngorn1 37927	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = dom dom 𝐻)
10		xpid11 5896	. . . . . . . 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 228	. . . . . . 7 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1^st ‘𝑅) × ran (1^st ‘𝑅)))
12		feq23 6669	. . . . . . 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 688	. . . . . 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 3296	. . . . . . . 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 3311	. . . . . . 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 3311	. . . . . 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 3296	. . . . . . 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 3312	. . . . . 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 1438	. . . . 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 2737	. . . 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 1343	. 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 6871	. . . 4 ⊢ (2^nd ‘𝑅) ∈ V
24		eleq1 2816	. . . 4 ⊢ (𝐻 = (2^nd ‘𝑅) → (𝐻 ∈ V ↔ (2^nd ‘𝑅) ∈ V))
25	23, 24	mpbiri 258	. . 3 ⊢ (𝐻 = (2^nd ‘𝑅) → 𝐻 ∈ V)
26		eqid 2729	. . . 4 ⊢ dom dom 𝐻 = dom dom 𝐻
27	26	ismndo1 37867	. . 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 234	1 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 Vcvv 3447 × cxp 5636 dom cdm 5638 ran crn 5639 ⟶wf 6507 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 AbelOpcablo 30473 MndOpcmndo 37860 RingOpscrngo 37888

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5251 ax-nul 5261 ax-pr 5387 ax-un 7711

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-fo 6517 df-fv 6519 df-ov 7390 df-1st 7968 df-2nd 7969 df-grpo 30422 df-ablo 30474 df-ass 37837 df-exid 37839 df-mgmOLD 37843 df-sgrOLD 37855 df-mndo 37861 df-rngo 37889

This theorem is referenced by: rngoidmlem 37930 rngo1cl 37933 isdrngo2 37952

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