Theorem spansncv

Description: Hilbert space has the covering property (using spans of singletons to represent atoms). Exercise 5 of Kalmbach p. 153. (Contributed by NM, 9-Jun-2004) (New usage is discouraged.)

	Ref	Expression
Assertion	spansncv	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) → 𝐵 = ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		psseq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ⊊ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ 𝐵 ) )
2		oveq1	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) )
3	2	sseq2d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐵 ⊆ ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ↔ 𝐵 ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) )
4	1 3	anbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ 𝐵 ∧ 𝐵 ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ) )
5	2	eqeq2d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( 𝐵 = ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ↔ 𝐵 = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) )
6	4 5	imbi12d	⊢ ( 𝐴 = if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) → ( ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) → 𝐵 = ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ 𝐵 ∧ 𝐵 ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) → 𝐵 = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ) )
7		psseq2	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ 𝐵 ↔ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ) )
8		sseq1	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( 𝐵 ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) )
9	7 8	anbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ 𝐵 ∧ 𝐵 ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ) )
10		eqeq1	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( 𝐵 = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) )
11	9 10	imbi12d	⊢ ( 𝐵 = if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) → ( ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ 𝐵 ∧ 𝐵 ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) → 𝐵 = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) → if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ) )
12		sneq	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → { 𝐶 } = { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } )
13	12	fveq2d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( span ‘ { 𝐶 } ) = ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) )
14	13	oveq2d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) )
15	14	sseq2d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) ) )
16	15	anbi2d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ↔ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) ) ) )
17	14	eqeq2d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ↔ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) ) )
18	16 17	imbi12d	⊢ ( 𝐶 = if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) → ( ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) → if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { 𝐶 } ) ) ) ↔ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) ) → if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) ) ) )
19		ifchhv	⊢ if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∈ Cℋ
20		ifchhv	⊢ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∈ Cℋ
21		ifhvhv0	⊢ if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) ∈ ℋ
22	19 20 21	spansncvi	⊢ ( ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ⊊ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ∧ if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) ⊆ ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) ) → if ( 𝐵 ∈ Cℋ , 𝐵 , ℋ ) = ( if ( 𝐴 ∈ Cℋ , 𝐴 , ℋ ) ∨ℋ ( span ‘ { if ( 𝐶 ∈ ℋ , 𝐶 , 0ℎ ) } ) ) )
23	6 11 18 22	dedth3h	⊢ ( ( 𝐴 ∈ Cℋ ∧ 𝐵 ∈ Cℋ ∧ 𝐶 ∈ ℋ ) → ( ( 𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) → 𝐵 = ( 𝐴 ∨ℋ ( span ‘ { 𝐶 } ) ) ) )