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Mirrors > Home > HSE Home > Th. List > norm3difi | 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, 16-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm3dif.1 | ⊢ 𝐴 ∈ ℋ |
norm3dif.2 | ⊢ 𝐵 ∈ ℋ |
norm3dif.3 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
norm3difi | ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | norm3dif.1 | . . . . 5 ⊢ 𝐴 ∈ ℋ | |
2 | norm3dif.2 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 2 | hvsubvali 28803 | . . . 4 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | norm3dif.3 | . . . . . . 7 ⊢ 𝐶 ∈ ℋ | |
5 | 1, 4 | hvsubvali 28803 | . . . . . 6 ⊢ (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) |
6 | 4, 2 | hvsubvali 28803 | . . . . . 6 ⊢ (𝐶 −ℎ 𝐵) = (𝐶 +ℎ (-1 ·ℎ 𝐵)) |
7 | 5, 6 | oveq12i 7147 | . . . . 5 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
8 | neg1cn 11739 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
9 | 8, 4 | hvmulcli 28797 | . . . . . 6 ⊢ (-1 ·ℎ 𝐶) ∈ ℋ |
10 | 8, 2 | hvmulcli 28797 | . . . . . . 7 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
11 | 4, 10 | hvaddcli 28801 | . . . . . 6 ⊢ (𝐶 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
12 | 1, 9, 11 | hvassi 28836 | . . . . 5 ⊢ ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) = (𝐴 +ℎ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵)))) |
13 | 9, 4, 10 | hvassi 28836 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
14 | 9, 4 | hvcomi 28802 | . . . . . . . . . 10 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
15 | 4, 4 | hvsubvali 28803 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
16 | hvsubid 28809 | . . . . . . . . . . 11 ⊢ (𝐶 ∈ ℋ → (𝐶 −ℎ 𝐶) = 0ℎ) | |
17 | 4, 16 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = 0ℎ |
18 | 14, 15, 17 | 3eqtr2i 2827 | . . . . . . . . 9 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = 0ℎ |
19 | 18 | oveq1i 7145 | . . . . . . . 8 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (0ℎ +ℎ (-1 ·ℎ 𝐵)) |
20 | ax-hv0cl 28786 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
21 | 20, 10 | hvcomi 28802 | . . . . . . . 8 ⊢ (0ℎ +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐵) +ℎ 0ℎ) |
22 | ax-hvaddid 28787 | . . . . . . . . 9 ⊢ ((-1 ·ℎ 𝐵) ∈ ℋ → ((-1 ·ℎ 𝐵) +ℎ 0ℎ) = (-1 ·ℎ 𝐵)) | |
23 | 10, 22 | ax-mp 5 | . . . . . . . 8 ⊢ ((-1 ·ℎ 𝐵) +ℎ 0ℎ) = (-1 ·ℎ 𝐵) |
24 | 19, 21, 23 | 3eqtri 2825 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (-1 ·ℎ 𝐵) |
25 | 13, 24 | eqtr3i 2823 | . . . . . 6 ⊢ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) = (-1 ·ℎ 𝐵) |
26 | 25 | oveq2i 7146 | . . . . 5 ⊢ (𝐴 +ℎ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵)))) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
27 | 7, 12, 26 | 3eqtri 2825 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
28 | 3, 27 | eqtr4i 2824 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) |
29 | 28 | fveq2i 6648 | . 2 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) |
30 | 1, 4 | hvsubcli 28804 | . . 3 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
31 | 4, 2 | hvsubcli 28804 | . . 3 ⊢ (𝐶 −ℎ 𝐵) ∈ ℋ |
32 | 30, 31 | norm-ii-i 28920 | . 2 ⊢ (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
33 | 29, 32 | eqbrtri 5051 | 1 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 1c1 10527 + caddc 10529 ≤ cle 10665 -cneg 10860 ℋchba 28702 +ℎ cva 28703 ·ℎ csm 28704 normℎcno 28706 0ℎc0v 28707 −ℎ cmv 28708 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-hfvadd 28783 ax-hvcom 28784 ax-hvass 28785 ax-hv0cl 28786 ax-hvaddid 28787 ax-hfvmul 28788 ax-hvmulid 28789 ax-hvmulass 28790 ax-hvdistr2 28792 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-hnorm 28751 df-hvsub 28754 |
This theorem is referenced by: norm3adifii 28931 norm3lem 28932 norm3dif 28933 |
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