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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 11105 | . . 3 ⊢ 0 ∈ ℂ | |
2 | 1 | fconst6 6729 | . 2 ⊢ ( ℋ × {0}): ℋ⟶ℂ |
3 | hvmulcl 29800 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
4 | hvaddcl 29799 | . . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) | |
5 | 3, 4 | sylan 580 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) |
6 | c0ex 11107 | . . . . . . 7 ⊢ 0 ∈ V | |
7 | 6 | fvconst2 7149 | . . . . . 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 7149 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
10 | 9 | oveq2d 7367 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 · (( ℋ × {0})‘𝑦)) = (𝑥 · 0)) |
11 | mul01 11292 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0) | |
12 | 10, 11 | sylan9eqr 2798 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (( ℋ × {0})‘𝑦)) = 0) |
13 | 6 | fvconst2 7149 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (( ℋ × {0})‘𝑧) = 0) |
14 | 12, 13 | oveqan12d 7370 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = (0 + 0)) |
15 | 00id 11288 | . . . . . 6 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | eqtrdi 2792 | . . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = 0) |
17 | 8, 16 | eqtr4d 2779 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
18 | 17 | 3impa 1110 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
19 | 18 | rgen3 3197 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) |
20 | ellnfn 30670 | . 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 709 | 1 ⊢ ( ℋ × {0}) ∈ LinFn |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 {csn 4584 × cxp 5629 ⟶wf 6489 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 0cc0 11009 + caddc 11012 · cmul 11014 ℋchba 29706 +ℎ cva 29707 ·ℎ csm 29708 LinFnclf 29741 |
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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-hilex 29786 ax-hfvadd 29787 ax-hfvmul 29792 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-po 5543 df-so 5544 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-ltxr 11152 df-lnfn 30635 |
This theorem is referenced by: nmfn0 30774 lnfn0 30834 lnfnmul 30835 nmbdfnlb 30837 nmcfnex 30840 nmcfnlb 30841 lnfncon 30843 riesz4 30851 riesz1 30852 |
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