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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 10032 | . . 3 ⊢ 0 ∈ ℂ | |
2 | 1 | fconst6 6095 | . 2 ⊢ ( ℋ × {0}): ℋ⟶ℂ |
3 | hvmulcl 27870 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 ·ℎ 𝑦) ∈ ℋ) | |
4 | hvaddcl 27869 | . . . . . . 7 ⊢ (((𝑥 ·ℎ 𝑦) ∈ ℋ ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) | |
5 | 3, 4 | sylan 488 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 ·ℎ 𝑦) +ℎ 𝑧) ∈ ℋ) |
6 | c0ex 10034 | . . . . . . 7 ⊢ 0 ∈ V | |
7 | 6 | fvconst2 6469 | . . . . . 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 6469 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℋ → (( ℋ × {0})‘𝑦) = 0) |
10 | 9 | oveq2d 6666 | . . . . . . . 8 ⊢ (𝑦 ∈ ℋ → (𝑥 · (( ℋ × {0})‘𝑦)) = (𝑥 · 0)) |
11 | mul01 10215 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ → (𝑥 · 0) = 0) | |
12 | 10, 11 | sylan9eqr 2678 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) → (𝑥 · (( ℋ × {0})‘𝑦)) = 0) |
13 | 6 | fvconst2 6469 | . . . . . . 7 ⊢ (𝑧 ∈ ℋ → (( ℋ × {0})‘𝑧) = 0) |
14 | 12, 13 | oveqan12d 6669 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = (0 + 0)) |
15 | 00id 10211 | . . . . . 6 ⊢ (0 + 0) = 0 | |
16 | 14, 15 | syl6eq 2672 | . . . . 5 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) = 0) |
17 | 8, 16 | eqtr4d 2659 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ) ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
18 | 17 | 3impa 1259 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℋ ∧ 𝑧 ∈ ℋ) → (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧))) |
19 | 18 | rgen3 2976 | . 2 ⊢ ∀𝑥 ∈ ℂ ∀𝑦 ∈ ℋ ∀𝑧 ∈ ℋ (( ℋ × {0})‘((𝑥 ·ℎ 𝑦) +ℎ 𝑧)) = ((𝑥 · (( ℋ × {0})‘𝑦)) + (( ℋ × {0})‘𝑧)) |
20 | ellnfn 28742 | . 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 955 | 1 ⊢ ( ℋ × {0}) ∈ LinFn |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {csn 4177 × cxp 5112 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 + caddc 9939 · cmul 9941 ℋchil 27776 +ℎ cva 27777 ·ℎ csm 27778 LinFnclf 27811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hilex 27856 ax-hfvadd 27857 ax-hfvmul 27862 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-lnfn 28707 |
This theorem is referenced by: nmfn0 28846 lnfn0 28906 lnfnmul 28907 nmbdfnlb 28909 nmcfnex 28912 nmcfnlb 28913 lnfncon 28915 riesz4 28923 riesz1 28924 |
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