Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > HSE Home > Th. List > homulcl | Structured version Visualization version GIF version |
Description: The scalar product of a Hilbert space operator is an operator. (Contributed by NM, 21-Feb-2006.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
homulcl | ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffvelrn 6851 | . . . . 5 ⊢ ((𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ) → (𝑇‘𝑥) ∈ ℋ) | |
2 | hvmulcl 28792 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇‘𝑥) ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝑥)) ∈ ℋ) | |
3 | 1, 2 | sylan2 594 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝑇: ℋ⟶ ℋ ∧ 𝑥 ∈ ℋ)) → (𝐴 ·ℎ (𝑇‘𝑥)) ∈ ℋ) |
4 | 3 | anassrs 470 | . . 3 ⊢ (((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) ∧ 𝑥 ∈ ℋ) → (𝐴 ·ℎ (𝑇‘𝑥)) ∈ ℋ) |
5 | 4 | fmpttd 6881 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))): ℋ⟶ ℋ) |
6 | hommval 29515 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇) = (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥)))) | |
7 | 6 | feq1d 6501 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → ((𝐴 ·op 𝑇): ℋ⟶ ℋ ↔ (𝑥 ∈ ℋ ↦ (𝐴 ·ℎ (𝑇‘𝑥))): ℋ⟶ ℋ)) |
8 | 5, 7 | mpbird 259 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝑇: ℋ⟶ ℋ) → (𝐴 ·op 𝑇): ℋ⟶ ℋ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℋchba 28698 ·ℎ csm 28700 ·op chot 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-hilex 28778 ax-hfvmul 28784 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-homul 29510 |
This theorem is referenced by: honegsubi 29575 homulid2 29579 homco1 29580 homulass 29581 hoadddi 29582 hoadddir 29583 hosubneg 29586 hosubdi 29587 honegsubdi 29589 honegsubdi2 29590 hosub4 29592 hosubsub4 29597 hosubeq0i 29605 nmopnegi 29744 homco2 29756 lnopmi 29779 hmopm 29800 nmophmi 29810 adjmul 29871 opsqrlem1 29919 opsqrlem6 29924 |
Copyright terms: Public domain | W3C validator |