Theorem chjatom

Description: The join of a closed subspace and an atom equals their subspace sum. Special case of remark in Kalmbach p. 65, stating that if A or B is finite-dimensional, then this equality holds. (Contributed by NM, 4-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	chjatom	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		atom1d	⊢ ( 𝐵 ∈ HAtoms ↔ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) )
2		spansnj	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ ℋ ) → ( 𝐴 +ℋ ( span ‘ { 𝑥 } ) ) = ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) )
3		oveq2	⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 +ℋ ( span ‘ { 𝑥 } ) ) )
4		oveq2	⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 ∨ℋ 𝐵 ) = ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) )
5	3 4	eqeq12d	⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ↔ ( 𝐴 +ℋ ( span ‘ { 𝑥 } ) ) = ( 𝐴 ∨ℋ ( span ‘ { 𝑥 } ) ) ) )
6	2 5	imbitrrid	⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ( 𝐴 ∈ Cℋ ∧ 𝑥 ∈ ℋ ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) )
7	6	expd	⊢ ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 ∈ Cℋ → ( 𝑥 ∈ ℋ → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) ) )
8	7	adantl	⊢ ( ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) → ( 𝐴 ∈ Cℋ → ( 𝑥 ∈ ℋ → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) ) )
9	8	com3l	⊢ ( 𝐴 ∈ Cℋ → ( 𝑥 ∈ ℋ → ( ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) ) )
10	9	rexlimdv	⊢ ( 𝐴 ∈ Cℋ → ( ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0ℎ ∧ 𝐵 = ( span ‘ { 𝑥 } ) ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) )
11	1 10	biimtrid	⊢ ( 𝐴 ∈ Cℋ → ( 𝐵 ∈ HAtoms → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) ) )
12	11	imp	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ HAtoms ) → ( 𝐴 +ℋ 𝐵 ) = ( 𝐴 ∨ℋ 𝐵 ) )