![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > normsubi | Structured version Visualization version GIF version |
Description: Negative doesn't change the norm of a Hilbert space vector. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normsub.1 | ⊢ 𝐴 ∈ ℋ |
normsub.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
normsubi | ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neg1cn 12264 | . . 3 ⊢ -1 ∈ ℂ | |
2 | normsub.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | normsub.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
4 | 2, 3 | hvsubcli 29861 | . . 3 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
5 | 1, 4 | norm-iii-i 29979 | . 2 ⊢ (normℎ‘(-1 ·ℎ (𝐵 −ℎ 𝐴))) = ((abs‘-1) · (normℎ‘(𝐵 −ℎ 𝐴))) |
6 | 2, 3 | hvnegdii 29902 | . . 3 ⊢ (-1 ·ℎ (𝐵 −ℎ 𝐴)) = (𝐴 −ℎ 𝐵) |
7 | 6 | fveq2i 6843 | . 2 ⊢ (normℎ‘(-1 ·ℎ (𝐵 −ℎ 𝐴))) = (normℎ‘(𝐴 −ℎ 𝐵)) |
8 | ax-1cn 11106 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | 8 | absnegi 15282 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
10 | abs1 15179 | . . . . 5 ⊢ (abs‘1) = 1 | |
11 | 9, 10 | eqtri 2764 | . . . 4 ⊢ (abs‘-1) = 1 |
12 | 11 | oveq1i 7364 | . . 3 ⊢ ((abs‘-1) · (normℎ‘(𝐵 −ℎ 𝐴))) = (1 · (normℎ‘(𝐵 −ℎ 𝐴))) |
13 | 4 | normcli 29971 | . . . . 5 ⊢ (normℎ‘(𝐵 −ℎ 𝐴)) ∈ ℝ |
14 | 13 | recni 11166 | . . . 4 ⊢ (normℎ‘(𝐵 −ℎ 𝐴)) ∈ ℂ |
15 | 14 | mulid2i 11157 | . . 3 ⊢ (1 · (normℎ‘(𝐵 −ℎ 𝐴))) = (normℎ‘(𝐵 −ℎ 𝐴)) |
16 | 12, 15 | eqtri 2764 | . 2 ⊢ ((abs‘-1) · (normℎ‘(𝐵 −ℎ 𝐴))) = (normℎ‘(𝐵 −ℎ 𝐴)) |
17 | 5, 7, 16 | 3eqtr3i 2772 | 1 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ‘cfv 6494 (class class class)co 7354 1c1 11049 · cmul 11053 -cneg 11383 abscabs 15116 ℋchba 29759 ·ℎ csm 29761 normℎcno 29763 −ℎ cmv 29765 |
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 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-pre-sup 11126 ax-hfvadd 29840 ax-hvcom 29841 ax-hv0cl 29843 ax-hfvmul 29845 ax-hvmulid 29846 ax-hvmulass 29847 ax-hvdistr1 29848 ax-hvmul0 29850 ax-hfi 29919 ax-his1 29922 ax-his3 29924 ax-his4 29925 |
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 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-sup 9375 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-div 11810 df-nn 12151 df-2 12213 df-3 12214 df-n0 12411 df-z 12497 df-uz 12761 df-rp 12913 df-seq 13904 df-exp 13965 df-cj 14981 df-re 14982 df-im 14983 df-sqrt 15117 df-abs 15118 df-hnorm 29808 df-hvsub 29811 |
This theorem is referenced by: normsub 29983 norm3adifii 29988 |
Copyright terms: Public domain | W3C validator |