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Mirrors > Home > HSE Home > Th. List > norm3lem | Structured version Visualization version GIF version |
Description: Lemma involving norm of differences in Hilbert space. (Contributed by NM, 18-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm3dif.1 | ⊢ 𝐴 ∈ ℋ |
norm3dif.2 | ⊢ 𝐵 ∈ ℋ |
norm3dif.3 | ⊢ 𝐶 ∈ ℋ |
norm3lem.4 | ⊢ 𝐷 ∈ ℝ |
Ref | Expression |
---|---|
norm3lem | ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | norm3dif.1 | . . . 4 ⊢ 𝐴 ∈ ℋ | |
2 | norm3dif.2 | . . . 4 ⊢ 𝐵 ∈ ℋ | |
3 | norm3dif.3 | . . . 4 ⊢ 𝐶 ∈ ℋ | |
4 | 1, 2, 3 | norm3difi 28580 | . . 3 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
5 | 1, 3 | hvsubcli 28454 | . . . . 5 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
6 | 5 | normcli 28564 | . . . 4 ⊢ (normℎ‘(𝐴 −ℎ 𝐶)) ∈ ℝ |
7 | 3, 2 | hvsubcli 28454 | . . . . 5 ⊢ (𝐶 −ℎ 𝐵) ∈ ℋ |
8 | 7 | normcli 28564 | . . . 4 ⊢ (normℎ‘(𝐶 −ℎ 𝐵)) ∈ ℝ |
9 | norm3lem.4 | . . . . 5 ⊢ 𝐷 ∈ ℝ | |
10 | 9 | rehalfcli 11635 | . . . 4 ⊢ (𝐷 / 2) ∈ ℝ |
11 | 6, 8, 10, 10 | lt2addi 10939 | . . 3 ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) < ((𝐷 / 2) + (𝐷 / 2))) |
12 | 1, 2 | hvsubcli 28454 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ |
13 | 12 | normcli 28564 | . . . 4 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ∈ ℝ |
14 | 6, 8 | readdcli 10394 | . . . 4 ⊢ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) ∈ ℝ |
15 | 10, 10 | readdcli 10394 | . . . 4 ⊢ ((𝐷 / 2) + (𝐷 / 2)) ∈ ℝ |
16 | 13, 14, 15 | lelttri 10505 | . . 3 ⊢ (((normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) ∧ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) < ((𝐷 / 2) + (𝐷 / 2))) → (normℎ‘(𝐴 −ℎ 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
17 | 4, 11, 16 | sylancr 581 | . 2 ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
18 | 10 | recni 10393 | . . . 4 ⊢ (𝐷 / 2) ∈ ℂ |
19 | 18 | 2timesi 11524 | . . 3 ⊢ (2 · (𝐷 / 2)) = ((𝐷 / 2) + (𝐷 / 2)) |
20 | 9 | recni 10393 | . . . 4 ⊢ 𝐷 ∈ ℂ |
21 | 2cn 11454 | . . . 4 ⊢ 2 ∈ ℂ | |
22 | 2ne0 11490 | . . . 4 ⊢ 2 ≠ 0 | |
23 | 20, 21, 22 | divcan2i 11120 | . . 3 ⊢ (2 · (𝐷 / 2)) = 𝐷 |
24 | 19, 23 | eqtr3i 2804 | . 2 ⊢ ((𝐷 / 2) + (𝐷 / 2)) = 𝐷 |
25 | 17, 24 | syl6breq 4929 | 1 ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 ℝcr 10273 + caddc 10277 · cmul 10279 < clt 10413 ≤ cle 10414 / cdiv 11034 2c2 11434 ℋchba 28352 normℎcno 28356 −ℎ cmv 28358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-hfvadd 28433 ax-hvcom 28434 ax-hvass 28435 ax-hv0cl 28436 ax-hvaddid 28437 ax-hfvmul 28438 ax-hvmulid 28439 ax-hvmulass 28440 ax-hvdistr2 28442 ax-hvmul0 28443 ax-hfi 28512 ax-his1 28515 ax-his2 28516 ax-his3 28517 ax-his4 28518 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-hnorm 28401 df-hvsub 28404 |
This theorem is referenced by: norm3lemt 28585 |
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