![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > HSE Home > Th. List > norm3adifii | Structured version Visualization version GIF version |
Description: Norm of differences around common element. Part of Lemma 3.6 of [Beran] p. 101. (Contributed by NM, 30-Sep-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm3dif.1 | ⊢ 𝐴 ∈ ℋ |
norm3dif.2 | ⊢ 𝐵 ∈ ℋ |
norm3dif.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
norm3adifii | ⊢ (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | norm3dif.1 | . . . . . . . 8 ⊢ 𝐴 ∈ ℋ | |
2 | norm3dif.3 | . . . . . . . 8 ⊢ 𝐶 ∈ ℋ | |
3 | 1, 2 | hvsubcli 28403 | . . . . . . 7 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
4 | 3 | normcli 28513 | . . . . . 6 ⊢ (normℎ‘(𝐴 −ℎ 𝐶)) ∈ ℝ |
5 | 4 | recni 10343 | . . . . 5 ⊢ (normℎ‘(𝐴 −ℎ 𝐶)) ∈ ℂ |
6 | norm3dif.2 | . . . . . . . 8 ⊢ 𝐵 ∈ ℋ | |
7 | 6, 2 | hvsubcli 28403 | . . . . . . 7 ⊢ (𝐵 −ℎ 𝐶) ∈ ℋ |
8 | 7 | normcli 28513 | . . . . . 6 ⊢ (normℎ‘(𝐵 −ℎ 𝐶)) ∈ ℝ |
9 | 8 | recni 10343 | . . . . 5 ⊢ (normℎ‘(𝐵 −ℎ 𝐶)) ∈ ℂ |
10 | 5, 9 | negsubdi2i 10659 | . . . 4 ⊢ -((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) = ((normℎ‘(𝐵 −ℎ 𝐶)) − (normℎ‘(𝐴 −ℎ 𝐶))) |
11 | 6, 2, 1 | norm3difi 28529 | . . . . . 6 ⊢ (normℎ‘(𝐵 −ℎ 𝐶)) ≤ ((normℎ‘(𝐵 −ℎ 𝐴)) + (normℎ‘(𝐴 −ℎ 𝐶))) |
12 | 6, 1 | normsubi 28523 | . . . . . . 7 ⊢ (normℎ‘(𝐵 −ℎ 𝐴)) = (normℎ‘(𝐴 −ℎ 𝐵)) |
13 | 12 | oveq1i 6888 | . . . . . 6 ⊢ ((normℎ‘(𝐵 −ℎ 𝐴)) + (normℎ‘(𝐴 −ℎ 𝐶))) = ((normℎ‘(𝐴 −ℎ 𝐵)) + (normℎ‘(𝐴 −ℎ 𝐶))) |
14 | 11, 13 | breqtri 4868 | . . . . 5 ⊢ (normℎ‘(𝐵 −ℎ 𝐶)) ≤ ((normℎ‘(𝐴 −ℎ 𝐵)) + (normℎ‘(𝐴 −ℎ 𝐶))) |
15 | 1, 6 | hvsubcli 28403 | . . . . . . 7 ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ |
16 | 15 | normcli 28513 | . . . . . 6 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ∈ ℝ |
17 | 8, 4, 16 | lesubaddi 10878 | . . . . 5 ⊢ (((normℎ‘(𝐵 −ℎ 𝐶)) − (normℎ‘(𝐴 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) ↔ (normℎ‘(𝐵 −ℎ 𝐶)) ≤ ((normℎ‘(𝐴 −ℎ 𝐵)) + (normℎ‘(𝐴 −ℎ 𝐶)))) |
18 | 14, 17 | mpbir 223 | . . . 4 ⊢ ((normℎ‘(𝐵 −ℎ 𝐶)) − (normℎ‘(𝐴 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) |
19 | 10, 18 | eqbrtri 4864 | . . 3 ⊢ -((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) |
20 | 4, 8 | resubcli 10635 | . . . 4 ⊢ ((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ∈ ℝ |
21 | 20, 16 | lenegcon1i 10872 | . . 3 ⊢ (-((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) ↔ -(normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) |
22 | 19, 21 | mpbi 222 | . 2 ⊢ -(normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) |
23 | 1, 2, 6 | norm3difi 28529 | . . 3 ⊢ (normℎ‘(𝐴 −ℎ 𝐶)) ≤ ((normℎ‘(𝐴 −ℎ 𝐵)) + (normℎ‘(𝐵 −ℎ 𝐶))) |
24 | 4, 8, 16 | lesubaddi 10878 | . . 3 ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) ↔ (normℎ‘(𝐴 −ℎ 𝐶)) ≤ ((normℎ‘(𝐴 −ℎ 𝐵)) + (normℎ‘(𝐵 −ℎ 𝐶)))) |
25 | 23, 24 | mpbir 223 | . 2 ⊢ ((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) |
26 | 20, 16 | abslei 14472 | . 2 ⊢ ((abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) ↔ (-(normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ∧ ((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)))) |
27 | 22, 25, 26 | mpbir2an 703 | 1 ⊢ (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2157 class class class wbr 4843 ‘cfv 6101 (class class class)co 6878 + caddc 10227 ≤ cle 10364 − cmin 10556 -cneg 10557 abscabs 14315 ℋchba 28301 normℎcno 28305 −ℎ cmv 28307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-pre-sup 10302 ax-hfvadd 28382 ax-hvcom 28383 ax-hvass 28384 ax-hv0cl 28385 ax-hvaddid 28386 ax-hfvmul 28387 ax-hvmulid 28388 ax-hvmulass 28389 ax-hvdistr1 28390 ax-hvdistr2 28391 ax-hvmul0 28392 ax-hfi 28461 ax-his1 28464 ax-his2 28465 ax-his3 28466 ax-his4 28467 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-div 10977 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-n0 11581 df-z 11667 df-uz 11931 df-rp 12075 df-seq 13056 df-exp 13115 df-cj 14180 df-re 14181 df-im 14182 df-sqrt 14316 df-abs 14317 df-hnorm 28350 df-hvsub 28353 |
This theorem is referenced by: norm3adifi 28535 |
Copyright terms: Public domain | W3C validator |