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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 31032 | . . . 4 ⊢ 0ℎ ∈ ℋ | |
2 | lnfnl.1 | . . . . . 6 ⊢ 𝑇 ∈ LinFn | |
3 | 2 | lnfnfi 32070 | . . . . 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 11522 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
7 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
8 | 2 | lnfnli 32069 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 0ℎ ∈ ℋ ∧ 0ℎ ∈ ℋ) → (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ))) |
9 | 7, 1, 1, 8 | mp3an 1460 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) |
10 | 7, 1 | hvmulcli 31043 | . . . . . . . . 9 ⊢ (1 ·ℎ 0ℎ) ∈ ℋ |
11 | ax-hvaddid 31033 | . . . . . . . . 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 31035 | . . . . . . . . 9 ⊢ (0ℎ ∈ ℋ → (1 ·ℎ 0ℎ) = 0ℎ) | |
14 | 1, 13 | ax-mp 5 | . . . . . . . 8 ⊢ (1 ·ℎ 0ℎ) = 0ℎ |
15 | 12, 14 | eqtri 2763 | . . . . . . 7 ⊢ ((1 ·ℎ 0ℎ) +ℎ 0ℎ) = 0ℎ |
16 | 15 | fveq2i 6910 | . . . . . 6 ⊢ (𝑇‘((1 ·ℎ 0ℎ) +ℎ 0ℎ)) = (𝑇‘0ℎ) |
17 | 9, 16 | eqtr3i 2765 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
18 | 5 | mullidi 11264 | . . . . . 6 ⊢ (1 · (𝑇‘0ℎ)) = (𝑇‘0ℎ) |
19 | 18 | oveq1i 7441 | . . . . 5 ⊢ ((1 · (𝑇‘0ℎ)) + (𝑇‘0ℎ)) = ((𝑇‘0ℎ) + (𝑇‘0ℎ)) |
20 | 17, 19 | eqtr3i 2765 | . . . 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 11578 | . . 3 ⊢ ((𝑇‘0ℎ) − (𝑇‘0ℎ)) = 0 |
23 | 21, 22 | eqtr3i 2765 | . 2 ⊢ (((𝑇‘0ℎ) + (𝑇‘0ℎ)) − (𝑇‘0ℎ)) = 0 |
24 | 6, 23 | eqtr3i 2765 | 1 ⊢ (𝑇‘0ℎ) = 0 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 − cmin 11490 ℋchba 30948 +ℎ cva 30949 ·ℎ csm 30950 0ℎc0v 30953 LinFnclf 30983 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-hilex 31028 ax-hv0cl 31032 ax-hvaddid 31033 ax-hfvmul 31034 ax-hvmulid 31035 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-ltxr 11298 df-sub 11492 df-lnfn 31877 |
This theorem is referenced by: lnfnmuli 32073 lnfn0 32076 nmbdfnlbi 32078 nmcfnexi 32080 nmcfnlbi 32081 nlelshi 32089 |
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