Theorem choc1

Description: The orthocomplement of the unit subspace is the zero subspace. Does not require Axiom of Choice. (Contributed by NM, 24-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	choc1	⊢ ( ⊥ ‘ ℋ ) = 0ℋ

Proof

Step	Hyp	Ref	Expression
1		helsh	⊢ ℋ ∈ Sℋ
2		shocel	⊢ ( ℋ ∈ Sℋ → ( 𝑥 ∈ ( ⊥ ‘ ℋ ) ↔ ( 𝑥 ∈ ℋ ∧ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ) ) )
3	1 2	ax-mp	⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) ↔ ( 𝑥 ∈ ℋ ∧ ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ) )
4	3	simprbi	⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 )
5		shocss	⊢ ( ℋ ∈ Sℋ → ( ⊥ ‘ ℋ ) ⊆ ℋ )
6	1 5	ax-mp	⊢ ( ⊥ ‘ ℋ ) ⊆ ℋ
7	6	sseli	⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → 𝑥 ∈ ℋ )
8		hial0	⊢ ( 𝑥 ∈ ℋ → ( ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ↔ 𝑥 = 0ℎ ) )
9	7 8	syl	⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → ( ∀ 𝑦 ∈ ℋ ( 𝑥 ·ih 𝑦 ) = 0 ↔ 𝑥 = 0ℎ ) )
10	4 9	mpbid	⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → 𝑥 = 0ℎ )
11		elch0	⊢ ( 𝑥 ∈ 0ℋ ↔ 𝑥 = 0ℎ )
12	10 11	sylibr	⊢ ( 𝑥 ∈ ( ⊥ ‘ ℋ ) → 𝑥 ∈ 0ℋ )
13	12	ssriv	⊢ ( ⊥ ‘ ℋ ) ⊆ 0ℋ
14		h0elsh	⊢ 0ℋ ∈ Sℋ
15		shococss	⊢ ( 0ℋ ∈ Sℋ → 0ℋ ⊆ ( ⊥ ‘ ( ⊥ ‘ 0ℋ ) ) )
16	14 15	ax-mp	⊢ 0ℋ ⊆ ( ⊥ ‘ ( ⊥ ‘ 0ℋ ) )
17		choc0	⊢ ( ⊥ ‘ 0ℋ ) = ℋ
18	17	fveq2i	⊢ ( ⊥ ‘ ( ⊥ ‘ 0ℋ ) ) = ( ⊥ ‘ ℋ )
19	16 18	sseqtri	⊢ 0ℋ ⊆ ( ⊥ ‘ ℋ )
20	13 19	eqssi	⊢ ( ⊥ ‘ ℋ ) = 0ℋ