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 ( ( 𝐴C𝐵 ∈ HAtoms ) → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) )

Proof

Step Hyp Ref Expression
1 atom1d ( 𝐵 ∈ HAtoms ↔ ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0𝐵 = ( span ‘ { 𝑥 } ) ) )
2 spansnj ( ( 𝐴C𝑥 ∈ ℋ ) → ( 𝐴 + ( span ‘ { 𝑥 } ) ) = ( 𝐴 ( span ‘ { 𝑥 } ) ) )
3 oveq2 ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 + 𝐵 ) = ( 𝐴 + ( span ‘ { 𝑥 } ) ) )
4 oveq2 ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴 𝐵 ) = ( 𝐴 ( span ‘ { 𝑥 } ) ) )
5 3 4 eqeq12d ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ↔ ( 𝐴 + ( span ‘ { 𝑥 } ) ) = ( 𝐴 ( span ‘ { 𝑥 } ) ) ) )
6 2 5 syl5ibr ( 𝐵 = ( span ‘ { 𝑥 } ) → ( ( 𝐴C𝑥 ∈ ℋ ) → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ) )
7 6 expd ( 𝐵 = ( span ‘ { 𝑥 } ) → ( 𝐴C → ( 𝑥 ∈ ℋ → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ) ) )
8 7 adantl ( ( 𝑥 ≠ 0𝐵 = ( span ‘ { 𝑥 } ) ) → ( 𝐴C → ( 𝑥 ∈ ℋ → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ) ) )
9 8 com3l ( 𝐴C → ( 𝑥 ∈ ℋ → ( ( 𝑥 ≠ 0𝐵 = ( span ‘ { 𝑥 } ) ) → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ) ) )
10 9 rexlimdv ( 𝐴C → ( ∃ 𝑥 ∈ ℋ ( 𝑥 ≠ 0𝐵 = ( span ‘ { 𝑥 } ) ) → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ) )
11 1 10 syl5bi ( 𝐴C → ( 𝐵 ∈ HAtoms → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) ) )
12 11 imp ( ( 𝐴C𝐵 ∈ HAtoms ) → ( 𝐴 + 𝐵 ) = ( 𝐴 𝐵 ) )