rngomndo - Mathbox for Jeff Madsen

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Theorem rngomndo 38434

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 2762	. . . 4 ⊢ (1^st ‘𝑅) = (1^st ‘𝑅)
2		unmnd.1	. . . 4 ⊢ 𝐻 = (2^nd ‘𝑅)
3		eqid 2762	. . . 4 ⊢ ran (1^st ‘𝑅) = ran (1^st ‘𝑅)
4	1, 2, 3	rngosm 38399	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran (1^st ‘𝑅) × ran (1^st ‘𝑅))⟶ran (1^st ‘𝑅))
5	1, 2, 3	rngoass 38405	. . . 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 3208	. . 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 38398	. . . 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 783	. . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran (1^st ‘𝑅)∀𝑦 ∈ ran (1^st ‘𝑅)((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦))
9	2, 1	rngorn1 38432	. . . 4 ⊢ (𝑅 ∈ RingOps → ran (1^st ‘𝑅) = dom dom 𝐻)
10		xpid11 5908	. . . . . . . 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 6672	. . . . . . 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 698	. . . . . 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 3317	. . . . . . . 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 3330	. . . . . . 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 3330	. . . . . 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 3317	. . . . . . 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 3331	. . . . . 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 1457	. . . . 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 2770	. . . 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 1356	. 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 6880	. . . 4 ⊢ (2^nd ‘𝑅) ∈ V
24		eleq1 2850	. . . 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 2762	. . . 4 ⊢ dom dom 𝐻 = dom dom 𝐻
27	26	ismndo1 38372	. . 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 399 ∧ w3a 1098 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ∃wrex 3086 Vcvv 3454 × cxp 5645 dom cdm 5647 ran crn 5648 ⟶wf 6517 ‘cfv 6521 (class class class)co 7396 1^st c1st 7968 2^nd c2nd 7969 AbelOpcablo 30747 MndOpcmndo 38365 RingOpscrngo 38393

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5246 ax-nul 5256 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4481 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5542 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-fo 6527 df-fv 6529 df-ov 7399 df-1st 7970 df-2nd 7971 df-grpo 30696 df-ablo 30748 df-ass 38342 df-exid 38344 df-mgmOLD 38348 df-sgrOLD 38360 df-mndo 38366 df-rngo 38394

This theorem is referenced by: rngoidmlem 38435 rngo1cl 38438 isdrngo2 38457

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