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Metamath Proof Explorer

Theorem hi2eq

Description: Lemma used to prove equality of vectors. (Contributed by NM, 16-Nov-1999) (New usage is discouraged.)

Ref Expression
Assertion hi2eq ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ↔ 𝐴 = 𝐵 ) )

Proof

Step Hyp Ref Expression
1 hvsubcl ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 −ℎ 𝐵 ) ∈ ℋ )
2 his2sub ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ ( 𝐴 −ℎ 𝐵 ) ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) − ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ) )
3 1 2 mpd3an3 ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) − ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ) )
4 3 eqeq1d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) − ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ) = 0 ) )
5 his6 ⊢ ( ( 𝐴 −ℎ 𝐵 ) ∈ ℋ → ( ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ ( 𝐴 −ℎ 𝐵 ) = 0ℎ ) )
6 1 5 syl ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( 𝐴 −ℎ 𝐵 ) ·ih ( 𝐴 −ℎ 𝐵 ) ) = 0 ↔ ( 𝐴 −ℎ 𝐵 ) = 0ℎ ) )
7 4 6 bitr3d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) − ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ) = 0 ↔ ( 𝐴 −ℎ 𝐵 ) = 0ℎ ) )
8 hicl ⊢ ( ( 𝐴 ∈ ℋ ∧ ( 𝐴 −ℎ 𝐵 ) ∈ ℋ ) → ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) ∈ ℂ )
9 1 8 syldan ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) ∈ ℂ )
10 simpr ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → 𝐵 ∈ ℋ )
11 hicl ⊢ ( ( 𝐵 ∈ ℋ ∧ ( 𝐴 −ℎ 𝐵 ) ∈ ℋ ) → ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ∈ ℂ )
12 10 1 11 syl2anc ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ∈ ℂ )
13 9 12 subeq0ad ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) − ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ) = 0 ↔ ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ) )
14 hvsubeq0 ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 −ℎ 𝐵 ) = 0ℎ ↔ 𝐴 = 𝐵 ) )
15 7 13 14 3bitr3d ⊢ ( ( 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ) → ( ( 𝐴 ·ih ( 𝐴 −ℎ 𝐵 ) ) = ( 𝐵 ·ih ( 𝐴 −ℎ 𝐵 ) ) ↔ 𝐴 = 𝐵 ) )