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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 9994 | . . 3 ⊢ 1 ∈ ℂ | |
2 | lnfnl.1 | . . . 4 ⊢ 𝑇 ∈ LinFn | |
3 | 2 | lnfnli 28899 | . . 3 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵))) |
4 | 1, 3 | mp3an1 1411 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵))) |
5 | ax-hvmulid 27863 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (1 ·ℎ 𝐴) = 𝐴) | |
6 | 5 | oveq1d 6665 | . . . 4 ⊢ (𝐴 ∈ ℋ → ((1 ·ℎ 𝐴) +ℎ 𝐵) = (𝐴 +ℎ 𝐵)) |
7 | 6 | fveq2d 6195 | . . 3 ⊢ (𝐴 ∈ ℋ → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
8 | 7 | adantr 481 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘((1 ·ℎ 𝐴) +ℎ 𝐵)) = (𝑇‘(𝐴 +ℎ 𝐵))) |
9 | 2 | lnfnfi 28900 | . . . . . 6 ⊢ 𝑇: ℋ⟶ℂ |
10 | 9 | ffvelrni 6358 | . . . . 5 ⊢ (𝐴 ∈ ℋ → (𝑇‘𝐴) ∈ ℂ) |
11 | 10 | mulid2d 10058 | . . . 4 ⊢ (𝐴 ∈ ℋ → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴)) |
12 | 11 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (1 · (𝑇‘𝐴)) = (𝑇‘𝐴)) |
13 | 12 | oveq1d 6665 | . 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((1 · (𝑇‘𝐴)) + (𝑇‘𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) |
14 | 4, 8, 13 | 3eqtr3d 2664 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝑇‘(𝐴 +ℎ 𝐵)) = ((𝑇‘𝐴) + (𝑇‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 1c1 9937 + caddc 9939 · cmul 9941 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 LinFnclf 27811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-mulcl 9998 ax-mulcom 10000 ax-mulass 10002 ax-distr 10003 ax-1rid 10006 ax-cnre 10009 ax-hilex 27856 ax-hvmulid 27863 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-map 7859 df-lnfn 28707 |
This theorem is referenced by: lnfnaddmuli 28904 nlelshi 28919 |
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