![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > lnfnaddi | Structured version Visualization version GIF version |
Description: Additive property of a linear Hilbert space functional. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfnl.1 | ⊢ 𝑇 ∈ LinFn |
Ref | Expression |
---|---|
lnfnaddi | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11106 | . . 3 ⊢ 1 ∈ ℂ | |
2 | lnfnl.1 | . . . 4 ⊢ 𝑇 ∈ LinFn | |
3 | 2 | lnfnli 30880 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵))) |
4 | 1, 3 | mp3an1 1448 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵))) |
5 | ax-hvmulid 29846 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
6 | 5 | fvoveq1d 7376 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
7 | 6 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
8 | 2 | lnfnfi 30881 | . . . . . 6 ⊢ 𝑇: ℋ⟶ℂ |
9 | 8 | ffvelcdmi 7031 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ) |
10 | 9 | mulid2d 11170 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴)) |
11 | 10 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴)) |
12 | 11 | oveq1d 7369 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) |
13 | 4, 7, 12 | 3eqtr3d 2784 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 ℂcc 11046 1c1 11049 + caddc 11051 · cmul 11053 ℋchba 29759 +ℎ cva 29760 ·ℎ csm 29761 LinFnclf 29794 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-mulcl 11110 ax-mulcom 11112 ax-mulass 11114 ax-distr 11115 ax-1rid 11118 ax-cnre 11121 ax-hilex 29839 ax-hvmulid 29846 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-fv 6502 df-ov 7357 df-oprab 7358 df-mpo 7359 df-map 8764 df-lnfn 30688 |
This theorem is referenced by: lnfnaddmuli 30885 nlelshi 30900 |
Copyright terms: Public domain | W3C validator |