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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 28724 | . . . 4 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | norm3dif.3 | . . . . . . 7 ⊢ 𝐶 ∈ ℋ | |
5 | 1, 4 | hvsubvali 28724 | . . . . . 6 ⊢ (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) |
6 | 4, 2 | hvsubvali 28724 | . . . . . 6 ⊢ (𝐶 −ℎ 𝐵) = (𝐶 +ℎ (-1 ·ℎ 𝐵)) |
7 | 5, 6 | oveq12i 7157 | . . . . 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 28718 | . . . . . 6 ⊢ (-1 ·ℎ 𝐶) ∈ ℋ |
10 | 8, 2 | hvmulcli 28718 | . . . . . . 7 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
11 | 4, 10 | hvaddcli 28722 | . . . . . 6 ⊢ (𝐶 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
12 | 1, 9, 11 | hvassi 28757 | . . . . 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 28757 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
14 | 9, 4 | hvcomi 28723 | . . . . . . . . . 10 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
15 | 4, 4 | hvsubvali 28724 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
16 | hvsubid 28730 | . . . . . . . . . . 11 ⊢ (𝐶 ∈ ℋ → (𝐶 −ℎ 𝐶) = 0ℎ) | |
17 | 4, 16 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = 0ℎ |
18 | 14, 15, 17 | 3eqtr2i 2847 | . . . . . . . . 9 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = 0ℎ |
19 | 18 | oveq1i 7155 | . . . . . . . 8 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (0ℎ +ℎ (-1 ·ℎ 𝐵)) |
20 | ax-hv0cl 28707 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
21 | 20, 10 | hvcomi 28723 | . . . . . . . 8 ⊢ (0ℎ +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐵) +ℎ 0ℎ) |
22 | ax-hvaddid 28708 | . . . . . . . . 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 2845 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (-1 ·ℎ 𝐵) |
25 | 13, 24 | eqtr3i 2843 | . . . . . 6 ⊢ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) = (-1 ·ℎ 𝐵) |
26 | 25 | oveq2i 7156 | . . . . 5 ⊢ (𝐴 +ℎ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵)))) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
27 | 7, 12, 26 | 3eqtri 2845 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
28 | 3, 27 | eqtr4i 2844 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) |
29 | 28 | fveq2i 6666 | . 2 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) |
30 | 1, 4 | hvsubcli 28725 | . . 3 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
31 | 4, 2 | hvsubcli 28725 | . . 3 ⊢ (𝐶 −ℎ 𝐵) ∈ ℋ |
32 | 30, 31 | norm-ii-i 28841 | . 2 ⊢ (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
33 | 29, 32 | eqbrtri 5078 | 1 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ‘cfv 6348 (class class class)co 7145 1c1 10526 + caddc 10528 ≤ cle 10664 -cneg 10859 ℋchba 28623 +ℎ cva 28624 ·ℎ csm 28625 normℎcno 28627 0ℎc0v 28628 −ℎ cmv 28629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvmulass 28711 ax-hvdistr2 28713 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 ax-his4 28789 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-hnorm 28672 df-hvsub 28675 |
This theorem is referenced by: norm3adifii 28852 norm3lem 28853 norm3dif 28854 |
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