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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 30119 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | lnfnl.1 | . . . . . 6 ⊢ 𝑇 ∈ LinFn | |
3 | 2 | lnfnfi 31157 | . . . . 5 ⊢ 𝑇: ℋ⟶ℂ |
4 | 3 | ffvelcdmi 7070 | . . . 4 ⊢ (0ℎ ∈ ℋ → (𝑇‘0ℎ) ∈ ℂ) |
5 | 1, 4 | ax-mp 5 | . . 3 ⊢ (𝑇‘0ℎ) ∈ ℂ |
6 | 5, 5 | pncan3oi 11458 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
7 | ax-1cn 11150 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 2 | lnfnli 31156 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ))) |
9 | 7, 1, 1, 8 | mp3an 1461 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) |
10 | 7, 1 | hvmulcli 30130 | . . . . . . . . 9 ⊢ (1 ·ℎ 0ℎ) ∈ ℋ |
11 | ax-hvaddid 30120 | . . . . . . . . 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 30122 | . . . . . . . . 9 ⊢ (0ℎ ∈ ℋ → (1 ·ℎ 0ℎ) = 0ℎ) | |
14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (1 ·ℎ 0ℎ) = 0ℎ |
15 | 12, 14 | eqtri 2759 | . . . . . . 7 ⊢ ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = 0ℎ |
16 | 15 | fveq2i 6881 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = (𝑇‘0ℎ) |
17 | 9, 16 | eqtr3i 2761 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
18 | 5 | mullidi 11201 | . . . . . 6 ⊢ (1 · (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
19 | 18 | oveq1i 7403 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
20 | 17, 19 | eqtr3i 2761 | . . . 4 ⊢ (𝑇‘0ℎ) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
21 | 20 | oveq1i 7403 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) |
22 | 5 | subidi 11513 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = 0 |
23 | 21, 22 | eqtr3i 2761 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = 0 |
24 | 6, 23 | eqtr3i 2761 | 1 ⊢ (𝑇‘0ℎ) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 0cc0 11092 1c1 11093 + caddc 11095 · cmul 11097 − cmin 11426 ℋchba 30035 +ℎ cva 30036 ·ℎ csm 30037 0ℎc0v 30040 LinFnclf 30070 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-hilex 30115 ax-hv0cl 30119 ax-hvaddid 30120 ax-hfvmul 30121 ax-hvmulid 30122 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-pnf 11232 df-mnf 11233 df-ltxr 11235 df-sub 11428 df-lnfn 30964 |
This theorem is referenced by: lnfnmuli 31160 lnfn0 31163 nmbdfnlbi 31165 nmcfnexi 31167 nmcfnlbi 31168 nlelshi 31176 |
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