![]() |
Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
|
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 29975 | . . 3 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
5 | 1, 3 | hvsubcli 29849 | . . . . 5 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
6 | 5 | normcli 29959 | . . . 4 ⊢ (normℎ‘(𝐴 −ℎ 𝐶)) ∈ ℝ |
7 | 3, 2 | hvsubcli 29849 | . . . . 5 ⊢ (𝐶 −ℎ 𝐵) ∈ ℋ |
8 | 7 | normcli 29959 | . . . 4 ⊢ (normℎ‘(𝐶 −ℎ 𝐵)) ∈ ℝ |
9 | norm3lem.4 | . . . . 5 ⊢ 𝐷 ∈ ℝ | |
10 | 9 | rehalfcli 12398 | . . . 4 ⊢ (𝐷 / 2) ∈ ℝ |
11 | 6, 8, 10, 10 | lt2addi 11713 | . . 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 29849 | . . . . 5 ⊢ (𝐴 −ℎ 𝐵) ∈ ℋ |
13 | 12 | normcli 29959 | . . . 4 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ∈ ℝ |
14 | 6, 8 | readdcli 11166 | . . . 4 ⊢ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) ∈ ℝ |
15 | 10, 10 | readdcli 11166 | . . . 4 ⊢ ((𝐷 / 2) + (𝐷 / 2)) ∈ ℝ |
16 | 13, 14, 15 | lelttri 11278 | . . 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 587 | . 2 ⊢ (((normℎ‘(𝐴 −ℎ 𝐶)) < (𝐷 / 2) ∧ (normℎ‘(𝐶 −ℎ 𝐵)) < (𝐷 / 2)) → (normℎ‘(𝐴 −ℎ 𝐵)) < ((𝐷 / 2) + (𝐷 / 2))) |
18 | 10 | recni 11165 | . . . 4 ⊢ (𝐷 / 2) ∈ ℂ |
19 | 18 | 2timesi 12287 | . . 3 ⊢ (2 · (𝐷 / 2)) = ((𝐷 / 2) + (𝐷 / 2)) |
20 | 9 | recni 11165 | . . . 4 ⊢ 𝐷 ∈ ℂ |
21 | 2cn 12224 | . . . 4 ⊢ 2 ∈ ℂ | |
22 | 2ne0 12253 | . . . 4 ⊢ 2 ≠ 0 | |
23 | 20, 21, 22 | divcan2i 11894 | . . 3 ⊢ (2 · (𝐷 / 2)) = 𝐷 |
24 | 19, 23 | eqtr3i 2766 | . 2 ⊢ ((𝐷 / 2) + (𝐷 / 2)) = 𝐷 |
25 | 17, 24 | breqtrdi 5144 | 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 396 ∈ wcel 2106 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 ℝcr 11046 + caddc 11050 · cmul 11052 < clt 11185 ≤ cle 11186 / cdiv 11808 2c2 12204 ℋchba 29747 normℎcno 29751 −ℎ cmv 29753 |
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 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-pre-sup 11125 ax-hfvadd 29828 ax-hvcom 29829 ax-hvass 29830 ax-hv0cl 29831 ax-hvaddid 29832 ax-hfvmul 29833 ax-hvmulid 29834 ax-hvmulass 29835 ax-hvdistr2 29837 ax-hvmul0 29838 ax-hfi 29907 ax-his1 29910 ax-his2 29911 ax-his3 29912 ax-his4 29913 |
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 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-2nd 7918 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9374 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-div 11809 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-n0 12410 df-z 12496 df-uz 12760 df-rp 12908 df-seq 13899 df-exp 13960 df-cj 14976 df-re 14977 df-im 14978 df-sqrt 15112 df-abs 15113 df-hnorm 29796 df-hvsub 29799 |
This theorem is referenced by: norm3lemt 29980 |
Copyright terms: Public domain | W3C validator |