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Metamath Proof Explorer

Theorem lnfnl

Description: Basic linearity property of a linear functional. (Contributed by NM, 11-Feb-2006) (New usage is discouraged.)

Ref Expression
Assertion lnfnl ( ( ( 𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ) ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ellnfn ⊢ ( 𝑇 ∈ LinFn ↔ ( 𝑇 : ℋ ⟶ ℂ ∧ ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℋ ∀ 𝑧 ∈ ℋ ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝑥 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) ) )
2 1 simprbi ⊢ ( 𝑇 ∈ LinFn → ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℋ ∀ 𝑧 ∈ ℋ ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝑥 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) )
3 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ·ℎ 𝑦 ) = ( 𝐴 ·ℎ 𝑦 ) )
4 3 fvoveq1d ⊢ ( 𝑥 = 𝐴 → ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) )
5 oveq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 · ( 𝑇 ‘ 𝑦 ) ) = ( 𝐴 · ( 𝑇 ‘ 𝑦 ) ) )
6 5 oveq1d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) )
7 4 6 eqeq12d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝑥 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) ↔ ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) ) )
8 oveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐴 ·ℎ 𝑦 ) = ( 𝐴 ·ℎ 𝐵 ) )
9 8 fvoveq1d ⊢ ( 𝑦 = 𝐵 → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝑧 ) ) )
10 fveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝑇 ‘ 𝑦 ) = ( 𝑇 ‘ 𝐵 ) )
11 10 oveq2d ⊢ ( 𝑦 = 𝐵 → ( 𝐴 · ( 𝑇 ‘ 𝑦 ) ) = ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) )
12 11 oveq1d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝑧 ) ) )
13 9 12 eqeq12d ⊢ ( 𝑦 = 𝐵 → ( ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) ↔ ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝑧 ) ) ) )
14 oveq2 ⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝑧 ) = ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) )
15 14 fveq2d ⊢ ( 𝑧 = 𝐶 → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝑧 ) ) = ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) )
16 fveq2 ⊢ ( 𝑧 = 𝐶 → ( 𝑇 ‘ 𝑧 ) = ( 𝑇 ‘ 𝐶 ) )
17 16 oveq2d ⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )
18 15 17 eqeq12d ⊢ ( 𝑧 = 𝐶 → ( ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝑧 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝑧 ) ) ↔ ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) ) )
19 7 13 18 rspc3v ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( ∀ 𝑥 ∈ ℂ ∀ 𝑦 ∈ ℋ ∀ 𝑧 ∈ ℋ ( 𝑇 ‘ ( ( 𝑥 ·ℎ 𝑦 ) +ℎ 𝑧 ) ) = ( ( 𝑥 · ( 𝑇 ‘ 𝑦 ) ) + ( 𝑇 ‘ 𝑧 ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) ) )
20 2 19 syl5 ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) → ( 𝑇 ∈ LinFn → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) ) )
21 20 3expb ⊢ ( ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ∈ LinFn → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) ) )
22 21 impcom ⊢ ( ( 𝑇 ∈ LinFn ∧ ( 𝐴 ∈ ℂ ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )
23 22 anassrs ⊢ ( ( ( 𝑇 ∈ LinFn ∧ 𝐴 ∈ ℂ ) ∧ ( 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ ) ) → ( 𝑇 ‘ ( ( 𝐴 ·ℎ 𝐵 ) +ℎ 𝐶 ) ) = ( ( 𝐴 · ( 𝑇 ‘ 𝐵 ) ) + ( 𝑇 ‘ 𝐶 ) ) )