Metamath Proof Explorer

Theorem honegsubi

Description: Relationship between Hilbert operator addition and subtraction. (Contributed by NM, 24-Aug-2006) (New usage is discouraged.)

Ref Expression
Hypotheses hodseq.2 ⊢ 𝑆 : ℋ ⟶ ℋ
hodseq.3 ⊢ 𝑇 : ℋ ⟶ ℋ
Assertion honegsubi ( 𝑆 +op ( - 1 ·op 𝑇 ) ) = ( 𝑆 −op 𝑇 )

Proof

Step Hyp Ref Expression
1 hodseq.2 ⊢ 𝑆 : ℋ ⟶ ℋ
2 hodseq.3 ⊢ 𝑇 : ℋ ⟶ ℋ
3 neg1cn ⊢ - 1 ∈ ℂ
4 homulcl ⊢ ( ( - 1 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ) → ( - 1 ·op 𝑇 ) : ℋ ⟶ ℋ )
5 3 2 4 mp2an ⊢ ( - 1 ·op 𝑇 ) : ℋ ⟶ ℋ
6 hosval ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ ( - 1 ·op 𝑇 ) : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( 𝑆 +op ( - 1 ·op 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( ( - 1 ·op 𝑇 ) ‘ 𝑥 ) ) )
7 1 5 6 mp3an12 ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 +op ( - 1 ·op 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( ( - 1 ·op 𝑇 ) ‘ 𝑥 ) ) )
8 1 ffvelcdmi ⊢ ( 𝑥 ∈ ℋ → ( 𝑆 ‘ 𝑥 ) ∈ ℋ )
9 2 ffvelcdmi ⊢ ( 𝑥 ∈ ℋ → ( 𝑇 ‘ 𝑥 ) ∈ ℋ )
10 hvsubval ⊢ ( ( ( 𝑆 ‘ 𝑥 ) ∈ ℋ ∧ ( 𝑇 ‘ 𝑥 ) ∈ ℋ ) → ( ( 𝑆 ‘ 𝑥 ) −ℎ ( 𝑇 ‘ 𝑥 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝑥 ) ) ) )
11 8 9 10 syl2anc ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 ‘ 𝑥 ) −ℎ ( 𝑇 ‘ 𝑥 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝑥 ) ) ) )
12 homval ⊢ ( ( - 1 ∈ ℂ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( - 1 ·op 𝑇 ) ‘ 𝑥 ) = ( - 1 ·ℎ ( 𝑇 ‘ 𝑥 ) ) )
13 3 2 12 mp3an12 ⊢ ( 𝑥 ∈ ℋ → ( ( - 1 ·op 𝑇 ) ‘ 𝑥 ) = ( - 1 ·ℎ ( 𝑇 ‘ 𝑥 ) ) )
14 13 oveq2d ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 ‘ 𝑥 ) +ℎ ( ( - 1 ·op 𝑇 ) ‘ 𝑥 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( - 1 ·ℎ ( 𝑇 ‘ 𝑥 ) ) ) )
15 11 14 eqtr4d ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 ‘ 𝑥 ) −ℎ ( 𝑇 ‘ 𝑥 ) ) = ( ( 𝑆 ‘ 𝑥 ) +ℎ ( ( - 1 ·op 𝑇 ) ‘ 𝑥 ) ) )
16 7 15 eqtr4d ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 +op ( - 1 ·op 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑥 ) −ℎ ( 𝑇 ‘ 𝑥 ) ) )
17 hodval ⊢ ( ( 𝑆 : ℋ ⟶ ℋ ∧ 𝑇 : ℋ ⟶ ℋ ∧ 𝑥 ∈ ℋ ) → ( ( 𝑆 −op 𝑇 ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑥 ) −ℎ ( 𝑇 ‘ 𝑥 ) ) )
18 1 2 17 mp3an12 ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 −op 𝑇 ) ‘ 𝑥 ) = ( ( 𝑆 ‘ 𝑥 ) −ℎ ( 𝑇 ‘ 𝑥 ) ) )
19 16 18 eqtr4d ⊢ ( 𝑥 ∈ ℋ → ( ( 𝑆 +op ( - 1 ·op 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 −op 𝑇 ) ‘ 𝑥 ) )
20 19 rgen ⊢ ∀ 𝑥 ∈ ℋ ( ( 𝑆 +op ( - 1 ·op 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 −op 𝑇 ) ‘ 𝑥 )
21 1 5 hoaddcli ⊢ ( 𝑆 +op ( - 1 ·op 𝑇 ) ) : ℋ ⟶ ℋ
22 1 2 hosubcli ⊢ ( 𝑆 −op 𝑇 ) : ℋ ⟶ ℋ
23 21 22 hoeqi ⊢ ( ∀ 𝑥 ∈ ℋ ( ( 𝑆 +op ( - 1 ·op 𝑇 ) ) ‘ 𝑥 ) = ( ( 𝑆 −op 𝑇 ) ‘ 𝑥 ) ↔ ( 𝑆 +op ( - 1 ·op 𝑇 ) ) = ( 𝑆 −op 𝑇 ) )
24 20 23 mpbi ⊢ ( 𝑆 +op ( - 1 ·op 𝑇 ) ) = ( 𝑆 −op 𝑇 )