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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 31022 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
| 2 | lnfnl.1 | . . . . . 6 ⊢ 𝑇 ∈ LinFn | |
| 3 | 2 | lnfnfi 32060 | . . . . 5 ⊢ 𝑇: ℋ⟶ℂ |
| 4 | 3 | ffvelcdmi 7103 | . . . 4 ⊢ (0ℎ ∈ ℋ → (𝑇‘0ℎ) ∈ ℂ) |
| 5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (𝑇‘0ℎ) ∈ ℂ |
| 6 | 5, 5 | pncan3oi 11524 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
| 7 | ax-1cn 11213 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
| 8 | 2 | lnfnli 32059 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ))) |
| 9 | 7, 1, 1, 8 | mp3an 1463 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) |
| 10 | 7, 1 | hvmulcli 31033 | . . . . . . . . 9 ⊢ (1 ·ℎ 0ℎ) ∈ ℋ |
| 11 | ax-hvaddid 31023 | . . . . . . . . 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 31025 | . . . . . . . . 9 ⊢ (0ℎ ∈ ℋ → (1 ·ℎ 0ℎ) = 0ℎ) | |
| 14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (1 ·ℎ 0ℎ) = 0ℎ |
| 15 | 12, 14 | eqtri 2765 | . . . . . . 7 ⊢ ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = 0ℎ |
| 16 | 15 | fveq2i 6909 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = (𝑇‘0ℎ) |
| 17 | 9, 16 | eqtr3i 2767 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
| 18 | 5 | mullidi 11266 | . . . . . 6 ⊢ (1 · (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
| 19 | 18 | oveq1i 7441 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
| 20 | 17, 19 | eqtr3i 2767 | . . . 4 ⊢ (𝑇‘0ℎ) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
| 21 | 20 | oveq1i 7441 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) |
| 22 | 5 | subidi 11580 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = 0 |
| 23 | 21, 22 | eqtr3i 2767 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = 0 |
| 24 | 6, 23 | eqtr3i 2767 | 1 ⊢ (𝑇‘0ℎ) = 0 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 − cmin 11492 ℋchba 30938 +ℎ cva 30939 ·ℎ csm 30940 0ℎc0v 30943 LinFnclf 30973 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-hilex 31018 ax-hv0cl 31022 ax-hvaddid 31023 ax-hfvmul 31024 ax-hvmulid 31025 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-ltxr 11300 df-sub 11494 df-lnfn 31867 |
| This theorem is referenced by: lnfnmuli 32063 lnfn0 32066 nmbdfnlbi 32068 nmcfnexi 32070 nmcfnlbi 32071 nlelshi 32079 |
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