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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 11001 | . . 3 ⊢ -1 ∈ ℂ | |
2 | normsub.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | normsub.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
4 | 2, 3 | hvsubcli 27262 | . . 3 ⊢ (𝐵 −ℎ 𝐴) ∈ ℋ |
5 | 1, 4 | norm-iii-i 27380 | . 2 ⊢ (normℎ‘(-1 ·ℎ (𝐵 −ℎ 𝐴))) = ((abs‘-1) · (normℎ‘(𝐵 −ℎ 𝐴))) |
6 | 2, 3 | hvnegdii 27303 | . . 3 ⊢ (-1 ·ℎ (𝐵 −ℎ 𝐴)) = (𝐴 −ℎ 𝐵) |
7 | 6 | fveq2i 6106 | . 2 ⊢ (normℎ‘(-1 ·ℎ (𝐵 −ℎ 𝐴))) = (normℎ‘(𝐴 −ℎ 𝐵)) |
8 | ax-1cn 9873 | . . . . . 6 ⊢ 1 ∈ ℂ | |
9 | 8 | absnegi 13987 | . . . . 5 ⊢ (abs‘-1) = (abs‘1) |
10 | abs1 13885 | . . . . 5 ⊢ (abs‘1) = 1 | |
11 | 9, 10 | eqtri 2632 | . . . 4 ⊢ (abs‘-1) = 1 |
12 | 11 | oveq1i 6559 | . . 3 ⊢ ((abs‘-1) · (normℎ‘(𝐵 −ℎ 𝐴))) = (1 · (normℎ‘(𝐵 −ℎ 𝐴))) |
13 | 4 | normcli 27372 | . . . . 5 ⊢ (normℎ‘(𝐵 −ℎ 𝐴)) ∈ ℝ |
14 | 13 | recni 9931 | . . . 4 ⊢ (normℎ‘(𝐵 −ℎ 𝐴)) ∈ ℂ |
15 | 14 | mulid2i 9922 | . . 3 ⊢ (1 · (normℎ‘(𝐵 −ℎ 𝐴))) = (normℎ‘(𝐵 −ℎ 𝐴)) |
16 | 12, 15 | eqtri 2632 | . 2 ⊢ ((abs‘-1) · (normℎ‘(𝐵 −ℎ 𝐴))) = (normℎ‘(𝐵 −ℎ 𝐴)) |
17 | 5, 7, 16 | 3eqtr3i 2640 | 1 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(𝐵 −ℎ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 1c1 9816 · cmul 9820 -cneg 10146 abscabs 13822 ℋchil 27160 ·ℎ csm 27162 normℎcno 27164 −ℎ cmv 27166 |
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-pre-mulgt0 9892 ax-pre-sup 9893 ax-hfvadd 27241 ax-hvcom 27242 ax-hv0cl 27244 ax-hfvmul 27246 ax-hvmulid 27247 ax-hvmulass 27248 ax-hvdistr1 27249 ax-hvmul0 27251 ax-hfi 27320 ax-his1 27323 ax-his3 27325 ax-his4 27326 |
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-reu 2903 df-rmo 2904 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-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 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-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-hnorm 27209 df-hvsub 27212 |
This theorem is referenced by: normsub 27384 norm3adifii 27389 |
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