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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 10621 | . . 3 ⊢ 0 ∈ ℂ | |
2 | 1 | fconst6 6562 | . 2 ⊢ ( ℋ × {0}): ℋ⟶ℂ |
3 | hvmulcl 28717 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
4 | hvaddcl 28716 | . . . . . . 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 10623 | . . . . . . 7 ⊢ 0 ∈ V | |
7 | 6 | fvconst2 6958 | . . . . . 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 6958 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
10 | 9 | oveq2d 7161 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 · (( ℋ × {0})‘𝑦)) = (𝑥 · 0)) |
11 | mul01 10807 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0) | |
12 | 10, 11 | sylan9eqr 2875 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (( ℋ × {0})‘𝑦)) = 0) |
13 | 6 | fvconst2 6958 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (( ℋ × {0})‘𝑧) = 0) |
14 | 12, 13 | oveqan12d 7164 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = (0 + 0)) |
15 | 00id 10803 | . . . . . 6 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | syl6eq 2869 | . . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = 0) |
17 | 8, 16 | eqtr4d 2856 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
18 | 17 | 3impa 1102 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
19 | 18 | rgen3 3201 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) |
20 | ellnfn 29587 | . 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 707 | 1 ⊢ ( ℋ × {0}) ∈ LinFn |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 + caddc 10528 · cmul 10530 ℋchba 28623 +ℎ cva 28624 ·ℎ csm 28625 LinFnclf 28658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-hilex 28703 ax-hfvadd 28704 ax-hfvmul 28709 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 df-lnfn 29552 |
This theorem is referenced by: nmfn0 29691 lnfn0 29751 lnfnmul 29752 nmbdfnlb 29754 nmcfnex 29757 nmcfnlb 29758 lnfncon 29760 riesz4 29768 riesz1 29769 |
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