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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 28800 | . . . 4 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | norm3dif.3 | . . . . . . 7 ⊢ 𝐶 ∈ ℋ | |
5 | 1, 4 | hvsubvali 28800 | . . . . . 6 ⊢ (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) |
6 | 4, 2 | hvsubvali 28800 | . . . . . 6 ⊢ (𝐶 −ℎ 𝐵) = (𝐶 +ℎ (-1 ·ℎ 𝐵)) |
7 | 5, 6 | oveq12i 7171 | . . . . 5 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
8 | neg1cn 11754 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
9 | 8, 4 | hvmulcli 28794 | . . . . . 6 ⊢ (-1 ·ℎ 𝐶) ∈ ℋ |
10 | 8, 2 | hvmulcli 28794 | . . . . . . 7 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
11 | 4, 10 | hvaddcli 28798 | . . . . . 6 ⊢ (𝐶 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
12 | 1, 9, 11 | hvassi 28833 | . . . . 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 28833 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
14 | 9, 4 | hvcomi 28799 | . . . . . . . . . 10 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
15 | 4, 4 | hvsubvali 28800 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
16 | hvsubid 28806 | . . . . . . . . . . 11 ⊢ (𝐶 ∈ ℋ → (𝐶 −ℎ 𝐶) = 0ℎ) | |
17 | 4, 16 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = 0ℎ |
18 | 14, 15, 17 | 3eqtr2i 2853 | . . . . . . . . 9 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = 0ℎ |
19 | 18 | oveq1i 7169 | . . . . . . . 8 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (0ℎ +ℎ (-1 ·ℎ 𝐵)) |
20 | ax-hv0cl 28783 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
21 | 20, 10 | hvcomi 28799 | . . . . . . . 8 ⊢ (0ℎ +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐵) +ℎ 0ℎ) |
22 | ax-hvaddid 28784 | . . . . . . . . 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 2851 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (-1 ·ℎ 𝐵) |
25 | 13, 24 | eqtr3i 2849 | . . . . . 6 ⊢ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) = (-1 ·ℎ 𝐵) |
26 | 25 | oveq2i 7170 | . . . . 5 ⊢ (𝐴 +ℎ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵)))) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
27 | 7, 12, 26 | 3eqtri 2851 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
28 | 3, 27 | eqtr4i 2850 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) |
29 | 28 | fveq2i 6676 | . 2 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) |
30 | 1, 4 | hvsubcli 28801 | . . 3 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
31 | 4, 2 | hvsubcli 28801 | . . 3 ⊢ (𝐶 −ℎ 𝐵) ∈ ℋ |
32 | 30, 31 | norm-ii-i 28917 | . 2 ⊢ (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
33 | 29, 32 | eqbrtri 5090 | 1 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 1c1 10541 + caddc 10543 ≤ cle 10679 -cneg 10874 ℋchba 28699 +ℎ cva 28700 ·ℎ csm 28701 normℎcno 28703 0ℎc0v 28704 −ℎ cmv 28705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-hfvadd 28780 ax-hvcom 28781 ax-hvass 28782 ax-hv0cl 28783 ax-hvaddid 28784 ax-hfvmul 28785 ax-hvmulid 28786 ax-hvmulass 28787 ax-hvdistr2 28789 ax-hvmul0 28790 ax-hfi 28859 ax-his1 28862 ax-his2 28863 ax-his3 28864 ax-his4 28865 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-hnorm 28748 df-hvsub 28751 |
This theorem is referenced by: norm3adifii 28928 norm3lem 28929 norm3dif 28930 |
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