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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 10597 | . . 3 ⊢ 1 ∈ ℂ | |
2 | lnopl.1 | . . . 4 ⊢ 𝑇 ∈ LinOp | |
3 | 2 | lnopli 29747 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 ·ℎ (𝑇‘𝐴)) +ℎ (𝑇‘𝐵))) |
4 | 1, 3 | mp3an1 1444 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 ·ℎ (𝑇‘𝐴)) +ℎ (𝑇‘𝐵))) |
5 | ax-hvmulid 28785 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
6 | 5 | fvoveq1d 7180 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
7 | 6 | adantr 483 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
8 | 2 | lnopfi 29748 | . . . . . 6 ⊢ 𝑇: ℋ⟶ ℋ |
9 | 8 | ffvelrni 6852 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℋ) |
10 | ax-hvmulid 28785 | . . . . 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 483 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 ·ℎ (𝑇‘𝐴)) = (𝑇‘𝐴)) |
13 | 12 | oveq1d 7173 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 ·ℎ (𝑇‘𝐴)) +ℎ (𝑇‘𝐵)) = ((𝑇‘𝐴) +ℎ (𝑇‘𝐵))) |
14 | 4, 7, 13 | 3eqtr3d 2866 | 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 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 1c1 10540 ℋchba 28698 +ℎ cva 28699 ·ℎ csm 28700 LinOpclo 28726 |
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-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-1cn 10597 ax-hilex 28778 ax-hvmulid 28785 |
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-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 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-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-map 8410 df-lnop 29620 |
This theorem is referenced by: lnopaddmuli 29752 lnophsi 29780 lnopeq0lem1 29784 lnophmlem2 29796 imaelshi 29837 cnlnadjlem2 29847 |
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