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Mirrors > Home > HSE Home > Th. List > lnopaddi | Structured version Visualization version GIF version |
Description: Additive property of a linear Hilbert space operator. (Contributed by NM, 11-May-2005.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnopl.1 | ⊢ 𝑇 ∈ LinOp |
Ref | Expression |
---|---|
lnopaddi | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) +ℎ (𝑇‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 10584 | . . 3 ⊢ 1 ∈ ℂ | |
2 | lnopl.1 | . . . 4 ⊢ 𝑇 ∈ LinOp | |
3 | 2 | lnopli 29751 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 ·ℎ (𝑇‘𝐴)) +ℎ (𝑇‘𝐵))) |
4 | 1, 3 | mp3an1 1445 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 ·ℎ (𝑇‘𝐴)) +ℎ (𝑇‘𝐵))) |
5 | ax-hvmulid 28789 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
6 | 5 | fvoveq1d 7157 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
7 | 6 | adantr 484 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
8 | 2 | lnopfi 29752 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ |
9 | 8 | ffvelrni 6827 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
10 | ax-hvmulid 28789 | . . . . 5 ⊢ ((𝑇‘𝐴) ∈ ℋ → (1 ·ℎ (𝑇‘𝐴)) = (𝑇‘𝐴)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ (𝑇‘𝐴)) = (𝑇‘𝐴)) |
12 | 11 | adantr 484 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 ·ℎ (𝑇‘𝐴)) = (𝑇‘𝐴)) |
13 | 12 | oveq1d 7150 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 ·ℎ (𝑇‘𝐴)) +ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +ℎ (𝑇‘𝐵))) |
14 | 4, 7, 13 | 3eqtr3d 2841 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) +ℎ (𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 1c1 10527 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 LinOpclo 28730 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-1cn 10584 ax-hilex 28782 ax-hvmulid 28789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-map 8391 df-lnop 29624 |
This theorem is referenced by: lnopaddmuli 29756 lnophsi 29784 lnopeq0lem1 29788 lnophmlem2 29800 imaelshi 29841 cnlnadjlem2 29851 |
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