Theorem gausslemma2dlem0d

Description: Auxiliary lemma 4 for gausslemma2d . (Contributed by AV, 9-Jul-2021)

	Ref	Expression
Hypotheses	gausslemma2dlem0.p	⊢ ( 𝜑 → 𝑃 ∈ ( ℙ ∖ { 2 } ) )
	gausslemma2dlem0.m	⊢ 𝑀 = ( ⌊ ‘ ( 𝑃 / 4 ) )
Assertion	gausslemma2dlem0d	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )

Proof

Step	Hyp	Ref	Expression
1		gausslemma2dlem0.p	⊢ ( 𝜑 → 𝑃 ∈ ( ℙ ∖ { 2 } ) )
2		gausslemma2dlem0.m	⊢ 𝑀 = ( ⌊ ‘ ( 𝑃 / 4 ) )
3	1	gausslemma2dlem0a	⊢ ( 𝜑 → 𝑃 ∈ ℕ )
4		nnre	⊢ ( 𝑃 ∈ ℕ → 𝑃 ∈ ℝ )
5		4re	⊢ 4 ∈ ℝ
6	5	a1i	⊢ ( 𝑃 ∈ ℕ → 4 ∈ ℝ )
7		4ne0	⊢ 4 ≠ 0
8	7	a1i	⊢ ( 𝑃 ∈ ℕ → 4 ≠ 0 )
9	4 6 8	redivcld	⊢ ( 𝑃 ∈ ℕ → ( 𝑃 / 4 ) ∈ ℝ )
10		nnnn0	⊢ ( 𝑃 ∈ ℕ → 𝑃 ∈ ℕ0 )
11	10	nn0ge0d	⊢ ( 𝑃 ∈ ℕ → 0 ≤ 𝑃 )
12		4pos	⊢ 0 < 4
13	5 12	pm3.2i	⊢ ( 4 ∈ ℝ ∧ 0 < 4 )
14	13	a1i	⊢ ( 𝑃 ∈ ℕ → ( 4 ∈ ℝ ∧ 0 < 4 ) )
15		divge0	⊢ ( ( ( 𝑃 ∈ ℝ ∧ 0 ≤ 𝑃 ) ∧ ( 4 ∈ ℝ ∧ 0 < 4 ) ) → 0 ≤ ( 𝑃 / 4 ) )
16	4 11 14 15	syl21anc	⊢ ( 𝑃 ∈ ℕ → 0 ≤ ( 𝑃 / 4 ) )
17	9 16	jca	⊢ ( 𝑃 ∈ ℕ → ( ( 𝑃 / 4 ) ∈ ℝ ∧ 0 ≤ ( 𝑃 / 4 ) ) )
18		flge0nn0	⊢ ( ( ( 𝑃 / 4 ) ∈ ℝ ∧ 0 ≤ ( 𝑃 / 4 ) ) → ( ⌊ ‘ ( 𝑃 / 4 ) ) ∈ ℕ0 )
19	3 17 18	3syl	⊢ ( 𝜑 → ( ⌊ ‘ ( 𝑃 / 4 ) ) ∈ ℕ0 )
20	2 19	eqeltrid	⊢ ( 𝜑 → 𝑀 ∈ ℕ0 )