nn0mnd - Mathbox for Alexander van der Vekens

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Theorem nn0mnd 46575

Description: The set of nonnegative integers under (complex) addition is a monoid. Example in [Lang] p. 6. Remark: 𝑀 could have also been written as (ℂ_fld ↾_s ℕ₀). (Contributed by AV, 27-Dec-2023.)

**Hypothesis**
Ref	Expression
nn0mnd.g	⊢ 𝑀 = {⟨(Base‘ndx), ℕ₀⟩, ⟨(+_g‘ndx), + ⟩}

**Assertion**
Ref	Expression
nn0mnd	⊢ 𝑀 ∈ Mnd

Proof of Theorem nn0mnd Dummy variables 𝑥 𝑦 𝑧 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nn0addcl 12503	. . . . 5 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → (𝑥 + 𝑦) ∈ ℕ₀)
2		nn0cn 12478	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ ℂ)
3		nn0cn 12478	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
4		nn0cn 12478	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ₀ → 𝑧 ∈ ℂ)
5	2, 3, 4	3anim123i 1151	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀ ∧ 𝑧 ∈ ℕ₀) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ))
6	5	3expa 1118	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) ∧ 𝑧 ∈ ℕ₀) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ))
7		addass 11193	. . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
8	6, 7	syl 17	. . . . . 6 ⊢ (((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) ∧ 𝑧 ∈ ℕ₀) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9	8	ralrimiva 3146	. . . . 5 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
10	1, 9	jca 512	. . . 4 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → ((𝑥 + 𝑦) ∈ ℕ₀ ∧ ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
11	10	rgen2 3197	. . 3 ⊢ ∀𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ ((𝑥 + 𝑦) ∈ ℕ₀ ∧ ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
12		c0ex 11204	. . . . 5 ⊢ 0 ∈ V
13		eleq1 2821	. . . . . 6 ⊢ (𝑒 = 0 → (𝑒 ∈ ℕ₀ ↔ 0 ∈ ℕ₀))
14		oveq1 7412	. . . . . . . . 9 ⊢ (𝑒 = 0 → (𝑒 + 𝑥) = (0 + 𝑥))
15	14	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑒 = 0 → ((𝑒 + 𝑥) = 𝑥 ↔ (0 + 𝑥) = 𝑥))
16		oveq2 7413	. . . . . . . . 9 ⊢ (𝑒 = 0 → (𝑥 + 𝑒) = (𝑥 + 0))
17	16	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑒 = 0 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 0) = 𝑥))
18	15, 17	anbi12d 631	. . . . . . 7 ⊢ (𝑒 = 0 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥)))
19	18	ralbidv 3177	. . . . . 6 ⊢ (𝑒 = 0 → (∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥)))
20	13, 19	anbi12d 631	. . . . 5 ⊢ (𝑒 = 0 → ((𝑒 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)) ↔ (0 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))))
21		0nn0 12483	. . . . . 6 ⊢ 0 ∈ ℕ₀
22	2	addlidd 11411	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → (0 + 𝑥) = 𝑥)
23	2	addridd 11410	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → (𝑥 + 0) = 𝑥)
24	22, 23	jca 512	. . . . . . 7 ⊢ (𝑥 ∈ ℕ₀ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
25	24	rgen 3063	. . . . . 6 ⊢ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥)
26	21, 25	pm3.2i 471	. . . . 5 ⊢ (0 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
27	12, 20, 26	ceqsexv2d 3528	. . . 4 ⊢ ∃𝑒(𝑒 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
28		df-rex 3071	. . . 4 ⊢ (∃𝑒 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∃𝑒(𝑒 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
29	27, 28	mpbir 230	. . 3 ⊢ ∃𝑒 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)
30	11, 29	pm3.2i 471	. 2 ⊢ (∀𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ ((𝑥 + 𝑦) ∈ ℕ₀ ∧ ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑒 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
31		nn0ex 12474	. . . 4 ⊢ ℕ₀ ∈ V
32		nn0mnd.g	. . . . 5 ⊢ 𝑀 = {⟨(Base‘ndx), ℕ₀⟩, ⟨(+_g‘ndx), + ⟩}
33	32	grpbase 17227	. . . 4 ⊢ (ℕ₀ ∈ V → ℕ₀ = (Base‘𝑀))
34	31, 33	ax-mp 5	. . 3 ⊢ ℕ₀ = (Base‘𝑀)
35		addex 12968	. . . 4 ⊢ + ∈ V
36	32	grpplusg 17229	. . . 4 ⊢ ( + ∈ V → + = (+_g‘𝑀))
37	35, 36	ax-mp 5	. . 3 ⊢ + = (+_g‘𝑀)
38	34, 37	ismnd 18624	. 2 ⊢ (𝑀 ∈ Mnd ↔ (∀𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ ((𝑥 + 𝑦) ∈ ℕ₀ ∧ ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑒 ∈ ℕ₀ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥)))
39	30, 38	mpbir 230	1 ⊢ 𝑀 ∈ Mnd

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 {cpr 4629 ⟨cop 4633 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 0cc0 11106 + caddc 11109 ℕ₀cn0 12468 ndxcnx 17122 Basecbs 17140 +_gcplusg 17193 Mndcmnd 18621

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-addf 11185

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mgm 18557 df-sgrp 18606 df-mnd 18622

This theorem is referenced by: (None)

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