Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > hfmval | Structured version Visualization version GIF version |
Description: Value of the scalar product with a Hilbert space functional. (Contributed by NM, 23-May-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hfmval | ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·fn 𝑇)‘𝐵) = (𝐴 · (𝑇‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hfmmval 29510 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → (𝐴 ·fn 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥)))) | |
2 | 1 | fveq1d 6667 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) → ((𝐴 ·fn 𝑇)‘𝐵) = ((𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥)))‘𝐵)) |
3 | fveq2 6665 | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝑇‘𝑥) = (𝑇‘𝐵)) | |
4 | 3 | oveq2d 7166 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝐴 · (𝑇‘𝑥)) = (𝐴 · (𝑇‘𝐵))) |
5 | eqid 2821 | . . . 4 ⊢ (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))) = (𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥))) | |
6 | ovex 7183 | . . . 4 ⊢ (𝐴 · (𝑇‘𝐵)) ∈ V | |
7 | 4, 5, 6 | fvmpt 6763 | . . 3 ⊢ (𝐵 ∈ ℋ → ((𝑥 ∈ ℋ ↦ (𝐴 · (𝑇‘𝑥)))‘𝐵) = (𝐴 · (𝑇‘𝐵))) |
8 | 2, 7 | sylan9eq 2876 | . 2 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ) ∧ 𝐵 ∈ ℋ) → ((𝐴 ·fn 𝑇)‘𝐵) = (𝐴 · (𝑇‘𝐵))) |
9 | 8 | 3impa 1106 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ℂ ∧ 𝐵 ∈ ℋ) → ((𝐴 ·fn 𝑇)‘𝐵) = (𝐴 · (𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ↦ cmpt 5139 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 · cmul 10536 ℋchba 28690 ·fn chft 28713 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-hilex 28770 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-map 8402 df-hfmul 29505 |
This theorem is referenced by: kbass2 29888 kbass3 29889 |
Copyright terms: Public domain | W3C validator |