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Mirrors > Home > HSE Home > Th. List > 0lnfn | Structured version Visualization version GIF version |
Description: The identically zero function is a linear Hilbert space functional. (Contributed by NM, 14-Feb-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
0lnfn | ⊢ ( ℋ × {0}) ∈ LinFn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0cn 9911 | . . 3 ⊢ 0 ∈ ℂ | |
2 | 1 | fconst6 6008 | . 2 ⊢ ( ℋ × {0}): ℋ⟶ℂ |
3 | hvmulcl 27254 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
4 | hvaddcl 27253 | . . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) | |
5 | 3, 4 | sylan 487 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) |
6 | c0ex 9913 | . . . . . . 7 ⊢ 0 ∈ V | |
7 | 6 | fvconst2 6374 | . . . . . 6 ⊢ (((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = 0) |
8 | 5, 7 | syl 17 | . . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = 0) |
9 | 6 | fvconst2 6374 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
10 | 9 | oveq2d 6565 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 · (( ℋ × {0})‘𝑦)) = (𝑥 · 0)) |
11 | mul01 10094 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0) | |
12 | 10, 11 | sylan9eqr 2666 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (( ℋ × {0})‘𝑦)) = 0) |
13 | 6 | fvconst2 6374 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (( ℋ × {0})‘𝑧) = 0) |
14 | 12, 13 | oveqan12d 6568 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = (0 + 0)) |
15 | 00id 10090 | . . . . . 6 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | syl6eq 2660 | . . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = 0) |
17 | 8, 16 | eqtr4d 2647 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
18 | 17 | 3impa 1251 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
19 | 18 | rgen3 2959 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) |
20 | ellnfn 28126 | . 2 ⊢ (( ℋ × {0}) ∈ LinFn ↔ (( ℋ × {0}): ℋ⟶ℂ ∧ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)))) | |
21 | 2, 19, 20 | mpbir2an 957 | 1 ⊢ ( ℋ × {0}) ∈ LinFn |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {csn 4125 × cxp 5036 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 + caddc 9818 · cmul 9820 ℋchil 27160 +ℎ cva 27161 ·ℎ csm 27162 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-hfvadd 27241 ax-hfvmul 27246 |
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-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-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-lnfn 28091 |
This theorem is referenced by: nmfn0 28230 lnfn0 28290 lnfnmul 28291 nmbdfnlb 28293 nmcfnex 28296 nmcfnlb 28297 lnfncon 28299 riesz4 28307 riesz1 28308 |
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