Theorem nnsgrpmgm

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

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

Proof

Step	Hyp	Ref	Expression
1		nnsgrp.m	⊢ 𝑀 = ( ℂfld ↾s ℕ )
2		1nn	⊢ 1 ∈ ℕ
3		nnaddcl	⊢ ( ( 𝑥 ∈ ℕ ∧ 𝑦 ∈ ℕ ) → ( 𝑥 + 𝑦 ) ∈ ℕ )
4	3	rgen2	⊢ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ( 𝑥 + 𝑦 ) ∈ ℕ
5		nnsscn	⊢ ℕ ⊆ ℂ
6	1	cnfldsrngbas	⊢ ( ℕ ⊆ ℂ → ℕ = ( Base ‘ 𝑀 ) )
7	5 6	ax-mp	⊢ ℕ = ( Base ‘ 𝑀 )
8		nnex	⊢ ℕ ∈ V
9	1	cnfldsrngadd	⊢ ( ℕ ∈ V → + = ( +g ‘ 𝑀 ) )
10	8 9	ax-mp	⊢ + = ( +g ‘ 𝑀 )
11	7 10	ismgmn0	⊢ ( 1 ∈ ℕ → ( 𝑀 ∈ Mgm ↔ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ( 𝑥 + 𝑦 ) ∈ ℕ ) )
12	4 11	mpbiri	⊢ ( 1 ∈ ℕ → 𝑀 ∈ Mgm )
13	2 12	ax-mp	⊢ 𝑀 ∈ Mgm