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Mirrors > Home > HSE Home > Th. List > lnfn0i | Structured version Visualization version GIF version |
Description: The value of a linear Hilbert space functional at zero is zero. Remark in [Beran] p. 99. (Contributed by NM, 11-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lnfnl.1 | ⊢ 𝑇 ∈ LinFn |
Ref | Expression |
---|---|
lnfn0i | ⊢ (𝑇‘0ℎ) = 0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hv0cl 27244 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | lnfnl.1 | . . . . . 6 ⊢ 𝑇 ∈ LinFn | |
3 | 2 | lnfnfi 28284 | . . . . 5 ⊢ 𝑇: ℋ⟶ℂ |
4 | 3 | ffvelrni 6266 | . . . 4 ⊢ (0ℎ ∈ ℋ → (𝑇‘0ℎ) ∈ ℂ) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (𝑇‘0ℎ) ∈ ℂ |
6 | 5, 5 | pncan3oi 10176 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
7 | ax-1cn 9873 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 2 | lnfnli 28283 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ))) |
9 | 7, 1, 1, 8 | mp3an 1416 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) |
10 | 7, 1 | hvmulcli 27255 | . . . . . . . . 9 ⊢ (1 ·ℎ 0ℎ) ∈ ℋ |
11 | ax-hvaddid 27245 | . . . . . . . . 9 ⊢ ((1 ·ℎ 0ℎ) ∈ ℋ → ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = (1 ·ℎ 0ℎ)) | |
12 | 10, 11 | ax-mp 5 | . . . . . . . 8 ⊢ ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = (1 ·ℎ 0ℎ) |
13 | ax-hvmulid 27247 | . . . . . . . . 9 ⊢ (0ℎ ∈ ℋ → (1 ·ℎ 0ℎ) = 0ℎ) | |
14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (1 ·ℎ 0ℎ) = 0ℎ |
15 | 12, 14 | eqtri 2632 | . . . . . . 7 ⊢ ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = 0ℎ |
16 | 15 | fveq2i 6106 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = (𝑇‘0ℎ) |
17 | 9, 16 | eqtr3i 2634 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
18 | 5 | mulid2i 9922 | . . . . . 6 ⊢ (1 · (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
19 | 18 | oveq1i 6559 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
20 | 17, 19 | eqtr3i 2634 | . . . 4 ⊢ (𝑇‘0ℎ) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
21 | 20 | oveq1i 6559 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) |
22 | 5 | subidi 10231 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = 0 |
23 | 21, 22 | eqtr3i 2634 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = 0 |
24 | 6, 23 | eqtr3i 2634 | 1 ⊢ (𝑇‘0ℎ) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 0ℎc0v 27165 LinFnclf 27195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-hilex 27240 ax-hv0cl 27244 ax-hvaddid 27245 ax-hfvmul 27246 ax-hvmulid 27247 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-ltxr 9958 df-sub 10147 df-lnfn 28091 |
This theorem is referenced by: lnfnmuli 28287 lnfn0 28290 nmbdfnlbi 28292 nmcfnexi 28294 nmcfnlbi 28295 nlelshi 28303 |
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