nn0mnd - Mathbox for Alexander van der Vekens

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Theorem nn0mnd 47110

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 12508	. . . . 5 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → (𝑥 + 𝑦) ∈ ℕ₀)
2		nn0cn 12483	. . . . . . . . 9 ⊢ (𝑥 ∈ ℕ₀ → 𝑥 ∈ ℂ)
3		nn0cn 12483	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ ℂ)
4		nn0cn 12483	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ₀ → 𝑧 ∈ ℂ)
5	2, 3, 4	3anim123i 1148	. . . . . . . 8 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀ ∧ 𝑧 ∈ ℕ₀) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ))
6	5	3expa 1115	. . . . . . 7 ⊢ (((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) ∧ 𝑧 ∈ ℕ₀) → (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ))
7		addass 11196	. . . . . . 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 3140	. . . . 5 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
10	1, 9	jca 511	. . . 4 ⊢ ((𝑥 ∈ ℕ₀ ∧ 𝑦 ∈ ℕ₀) → ((𝑥 + 𝑦) ∈ ℕ₀ ∧ ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))))
11	10	rgen2 3191	. . 3 ⊢ ∀𝑥 ∈ ℕ₀ ∀𝑦 ∈ ℕ₀ ((𝑥 + 𝑦) ∈ ℕ₀ ∧ ∀𝑧 ∈ ℕ₀ ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
12		c0ex 11209	. . . . 5 ⊢ 0 ∈ V
13		eleq1 2815	. . . . . 6 ⊢ (𝑒 = 0 → (𝑒 ∈ ℕ₀ ↔ 0 ∈ ℕ₀))
14		oveq1 7411	. . . . . . . . 9 ⊢ (𝑒 = 0 → (𝑒 + 𝑥) = (0 + 𝑥))
15	14	eqeq1d 2728	. . . . . . . 8 ⊢ (𝑒 = 0 → ((𝑒 + 𝑥) = 𝑥 ↔ (0 + 𝑥) = 𝑥))
16		oveq2 7412	. . . . . . . . 9 ⊢ (𝑒 = 0 → (𝑥 + 𝑒) = (𝑥 + 0))
17	16	eqeq1d 2728	. . . . . . . 8 ⊢ (𝑒 = 0 → ((𝑥 + 𝑒) = 𝑥 ↔ (𝑥 + 0) = 𝑥))
18	15, 17	anbi12d 630	. . . . . . 7 ⊢ (𝑒 = 0 → (((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥)))
19	18	ralbidv 3171	. . . . . 6 ⊢ (𝑒 = 0 → (∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥) ↔ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥)))
20	13, 19	anbi12d 630	. . . . 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 12488	. . . . . 6 ⊢ 0 ∈ ℕ₀
22	2	addlidd 11416	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → (0 + 𝑥) = 𝑥)
23	2	addridd 11415	. . . . . . . 8 ⊢ (𝑥 ∈ ℕ₀ → (𝑥 + 0) = 𝑥)
24	22, 23	jca 511	. . . . . . 7 ⊢ (𝑥 ∈ ℕ₀ → ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
25	24	rgen 3057	. . . . . 6 ⊢ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥)
26	21, 25	pm3.2i 470	. . . . 5 ⊢ (0 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((0 + 𝑥) = 𝑥 ∧ (𝑥 + 0) = 𝑥))
27	12, 20, 26	ceqsexv2d 3523	. . . 4 ⊢ ∃𝑒(𝑒 ∈ ℕ₀ ∧ ∀𝑥 ∈ ℕ₀ ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
28		df-rex 3065	. . . 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 470	. 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 12479	. . . 4 ⊢ ℕ₀ ∈ V
32		nn0mnd.g	. . . . 5 ⊢ 𝑀 = {⟨(Base‘ndx), ℕ₀⟩, ⟨(+_g‘ndx), + ⟩}
33	32	grpbase 17238	. . . 4 ⊢ (ℕ₀ ∈ V → ℕ₀ = (Base‘𝑀))
34	31, 33	ax-mp 5	. . 3 ⊢ ℕ₀ = (Base‘𝑀)
35		addex 12974	. . . 4 ⊢ + ∈ V
36	32	grpplusg 17240	. . . 4 ⊢ ( + ∈ V → + = (+_g‘𝑀))
37	35, 36	ax-mp 5	. . 3 ⊢ + = (+_g‘𝑀)
38	34, 37	ismnd 18668	. 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 395 ∧ w3a 1084 = wceq 1533 ∃wex 1773 ∈ wcel 2098 ∀wral 3055 ∃wrex 3064 Vcvv 3468 {cpr 4625 ⟨cop 4629 ‘cfv 6536 (class class class)co 7404 ℂcc 11107 0cc0 11109 + caddc 11112 ℕ₀cn0 12473 ndxcnx 17133 Basecbs 17151 +_gcplusg 17204 Mndcmnd 18665

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-addf 11188

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-2 12276 df-n0 12474 df-z 12560 df-uz 12824 df-fz 13488 df-struct 17087 df-slot 17122 df-ndx 17134 df-base 17152 df-plusg 17217 df-mgm 18571 df-sgrp 18650 df-mnd 18666

This theorem is referenced by: (None)

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