Theorem h1da

Description: A 1-dimensional subspace is an atom. (Contributed by NM, 22-Jul-2001) (New usage is discouraged.)

	Ref	Expression
Assertion	h1da	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ HAtoms )

Proof

Step	Hyp	Ref	Expression
1		snssi	⊢ ( 𝐴 ∈ ℋ → { 𝐴 } ⊆ ℋ )
2		occl	⊢ ( { 𝐴 } ⊆ ℋ → ( ⊥ ‘ { 𝐴 } ) ∈ Cℋ )
3		choccl	⊢ ( ( ⊥ ‘ { 𝐴 } ) ∈ Cℋ → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ Cℋ )
4	1 2 3	3syl	⊢ ( 𝐴 ∈ ℋ → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ Cℋ )
5	4	adantr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ Cℋ )
6		h1dn0	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ≠ 0ℋ )
7		h1datom	⊢ ( ( 𝑥 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( 𝑥 ⊆ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) → ( 𝑥 = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∨ 𝑥 = 0ℋ ) ) )
8	7	expcom	⊢ ( 𝐴 ∈ ℋ → ( 𝑥 ∈ Cℋ → ( 𝑥 ⊆ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) → ( 𝑥 = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∨ 𝑥 = 0ℋ ) ) ) )
9	8	ralrimiv	⊢ ( 𝐴 ∈ ℋ → ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) → ( 𝑥 = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∨ 𝑥 = 0ℋ ) ) )
10	9	adantr	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) → ( 𝑥 = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∨ 𝑥 = 0ℋ ) ) )
11	6 10	jca	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ≠ 0ℋ ∧ ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) → ( 𝑥 = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∨ 𝑥 = 0ℋ ) ) ) )
12		elat2	⊢ ( ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ HAtoms ↔ ( ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ Cℋ ∧ ( ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ≠ 0ℋ ∧ ∀ 𝑥 ∈ Cℋ ( 𝑥 ⊆ ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) → ( 𝑥 = ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∨ 𝑥 = 0ℋ ) ) ) ) )
13	5 11 12	sylanbrc	⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐴 ≠ 0ℎ ) → ( ⊥ ‘ ( ⊥ ‘ { 𝐴 } ) ) ∈ HAtoms )