Theorem nnsgrp

Description: The structure of positive integers together with the addition of complex numbers is a semigroup. (Contributed by AV, 4-Feb-2020)

	Ref	Expression
Hypothesis	nnsgrp.m	⊢ 𝑀 = ( ℂfld ↾s ℕ )
Assertion	nnsgrp	⊢ 𝑀 ∈ Smgrp

Proof

Step	Hyp	Ref	Expression
1		nnsgrp.m	⊢ 𝑀 = ( ℂfld ↾s ℕ )
2	1	nnsgrpmgm	⊢ 𝑀 ∈ Mgm
3		nncn	⊢ ( 𝑥 ∈ ℕ → 𝑥 ∈ ℂ )
4		nncn	⊢ ( 𝑦 ∈ ℕ → 𝑦 ∈ ℂ )
5		nncn	⊢ ( 𝑧 ∈ ℕ → 𝑧 ∈ ℂ )
6		addass	⊢ ( ( 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
7	3 4 5 6	syl3an	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
8	7	3expia	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑧 ∈ ℕ → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) )
9	8	ralrimiv	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ∀ 𝑧 ∈ ℕ ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) )
10	9	rgen2	⊢ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) )
11		nnsscn	⊢ ℕ ⊆ ℂ
12	1	cnfldsrngbas	⊢ ( ℕ ⊆ ℂ → ℕ = ( Base ‘ 𝑀 ) )
13	11 12	ax-mp	⊢ ℕ = ( Base ‘ 𝑀 )
14		nnex	⊢ ℕ ∈ V
15	1	cnfldsrngadd	⊢ ( ℕ ∈ V → + = ( +g ‘ 𝑀 ) )
16	14 15	ax-mp	⊢ + = ( +g ‘ 𝑀 )
17	13 16	issgrp	⊢ ( 𝑀 ∈ Smgrp ↔ ( 𝑀 ∈ Mgm ∧ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) )
18	2 10 17	mpbir2an	⊢ 𝑀 ∈ Smgrp