rngomndo - Mathbox for Jeff Madsen

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Theorem rngomndo 38302

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 2739	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		unmnd.1	. . . 4 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2739	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4	1, 2, 3	rngosm 38267	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅))
5	1, 2, 3	rngoass 38273	. . . 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 3185	. . 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 38266	. . . 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 779	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
9	2, 1	rngorn1 38300	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = dom dom 𝐻)
10		xpid11 5874	. . . . . . . 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 229	. . . . . . 7 ⊢ (dom dom 𝐻 = ran (1^st ‘𝑅) → (dom dom 𝐻 × dom dom 𝐻) = (ran (1^st ‘𝑅) × ran (1^st ‘𝑅)))
12		feq23 6636	. . . . . . 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 694	. . . . . 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 3294	. . . . . . . 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 3307	. . . . . . 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 3307	. . . . . 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 3294	. . . . . . 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 3308	. . . . . 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 1444	. . . . 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 2747	. . . 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 1349	. 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 6840	. . . 4 ⊢ (2^nd ‘𝑅) ∈ V
24		eleq1 2827	. . . 4 ⊢ (𝐻 = (2^nd ‘𝑅) → (𝐻 ∈ V ↔ (2^nd ‘𝑅) ∈ V))
25	23, 24	mpbiri 259	. . 3 ⊢ (𝐻 = (2^nd ‘𝑅) → 𝐻 ∈ V)
26		eqid 2739	. . . 4 ⊢ dom dom 𝐻 = dom dom 𝐻
27	26	ismndo1 38240	. . 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 235	1 ⊢ (𝑅 ∈ RingOps → 𝐻 ∈ MndOp)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∃wrex 3063 Vcvv 3431 × cxp 5616 dom cdm 5618 ran crn 5619 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 1^st c1st 7929 2^nd c2nd 7930 AbelOpcablo 30633 MndOpcmndo 38233 RingOpscrngo 38261

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-fo 6491 df-fv 6493 df-ov 7359 df-1st 7931 df-2nd 7932 df-grpo 30582 df-ablo 30634 df-ass 38210 df-exid 38212 df-mgmOLD 38216 df-sgrOLD 38228 df-mndo 38234 df-rngo 38262

This theorem is referenced by: rngoidmlem 38303 rngo1cl 38306 isdrngo2 38325

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