rngomndo - Mathbox for Jeff Madsen

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Theorem rngomndo 35215

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 2823	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		unmnd.1	. . . 4 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2823	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4	1, 2, 3	rngosm 35180	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅))
5	1, 2, 3	rngoass 35186	. . . 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 3194	. . 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 35179	. . . 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 772	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
9	2, 1	rngorn1 35213	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = dom dom 𝐻)
10		xpid11 5804	. . . . . . . 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 230	. . . . . . 7 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1^st ‘𝑅) × ran (1^st ‘𝑅)))
12		feq23 6500	. . . . . . 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 686	. . . . . 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 3407	. . . . . . . 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 3405	. . . . . . 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 3405	. . . . . 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 3407	. . . . . . 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 3406	. . . . . 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 1432	. . . . 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 2831	. . . 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 1338	. 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 6685	. . . 4 ⊢ (2^nd ‘𝑅) ∈ V
24		eleq1 2902	. . . 4 ⊢ (𝐻 = (2^nd ‘𝑅) → (𝐻 ∈ V ↔ (2^nd ‘𝑅) ∈ V))
25	23, 24	mpbiri 260	. . 3 ⊢ (𝐻 = (2^nd ‘𝑅) → 𝐻 ∈ V)
26		eqid 2823	. . . 4 ⊢ dom dom 𝐻 = dom dom 𝐻
27	26	ismndo1 35153	. . 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 236	1 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 Vcvv 3496 × cxp 5555 dom cdm 5557 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 AbelOpcablo 28323 MndOpcmndo 35146 RingOpscrngo 35174

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-grpo 28272 df-ablo 28324 df-ass 35123 df-exid 35125 df-mgmOLD 35129 df-sgrOLD 35141 df-mndo 35147 df-rngo 35175

This theorem is referenced by: rngoidmlem 35216 rngo1cl 35219 isdrngo2 35238

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