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Metamath Proof Explorer

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 ) )