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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 28786 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | lnfnl.1 | . . . . . 6 ⊢ 𝑇 ∈ LinFn | |
3 | 2 | lnfnfi 29824 | . . . . 5 ⊢ 𝑇: ℋ⟶ℂ |
4 | 3 | ffvelrni 6827 | . . . 4 ⊢ (0ℎ ∈ ℋ → (𝑇‘0ℎ) ∈ ℂ) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (𝑇‘0ℎ) ∈ ℂ |
6 | 5, 5 | pncan3oi 10891 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
7 | ax-1cn 10584 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 2 | lnfnli 29823 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ))) |
9 | 7, 1, 1, 8 | mp3an 1458 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) |
10 | 7, 1 | hvmulcli 28797 | . . . . . . . . 9 ⊢ (1 ·ℎ 0ℎ) ∈ ℋ |
11 | ax-hvaddid 28787 | . . . . . . . . 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 28789 | . . . . . . . . 9 ⊢ (0ℎ ∈ ℋ → (1 ·ℎ 0ℎ) = 0ℎ) | |
14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (1 ·ℎ 0ℎ) = 0ℎ |
15 | 12, 14 | eqtri 2821 | . . . . . . 7 ⊢ ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = 0ℎ |
16 | 15 | fveq2i 6648 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = (𝑇‘0ℎ) |
17 | 9, 16 | eqtr3i 2823 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
18 | 5 | mulid2i 10635 | . . . . . 6 ⊢ (1 · (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
19 | 18 | oveq1i 7145 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
20 | 17, 19 | eqtr3i 2823 | . . . 4 ⊢ (𝑇‘0ℎ) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
21 | 20 | oveq1i 7145 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) |
22 | 5 | subidi 10946 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = 0 |
23 | 21, 22 | eqtr3i 2823 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = 0 |
24 | 6, 23 | eqtr3i 2823 | 1 ⊢ (𝑇‘0ℎ) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 − cmin 10859 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 0ℎc0v 28707 LinFnclf 28737 |
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-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-hilex 28782 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 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-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 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-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-sub 10861 df-lnfn 29631 |
This theorem is referenced by: lnfnmuli 29827 lnfn0 29830 nmbdfnlbi 29832 nmcfnexi 29834 nmcfnlbi 29835 nlelshi 29843 |
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