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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 28592 | . . . 4 ⊢ (𝐴 −ℎ 𝐵) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
4 | norm3dif.3 | . . . . . . 7 ⊢ 𝐶 ∈ ℋ | |
5 | 1, 4 | hvsubvali 28592 | . . . . . 6 ⊢ (𝐴 −ℎ 𝐶) = (𝐴 +ℎ (-1 ·ℎ 𝐶)) |
6 | 4, 2 | hvsubvali 28592 | . . . . . 6 ⊢ (𝐶 −ℎ 𝐵) = (𝐶 +ℎ (-1 ·ℎ 𝐵)) |
7 | 5, 6 | oveq12i 6987 | . . . . 5 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = ((𝐴 +ℎ (-1 ·ℎ 𝐶)) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
8 | neg1cn 11560 | . . . . . . 7 ⊢ -1 ∈ ℂ | |
9 | 8, 4 | hvmulcli 28586 | . . . . . 6 ⊢ (-1 ·ℎ 𝐶) ∈ ℋ |
10 | 8, 2 | hvmulcli 28586 | . . . . . . 7 ⊢ (-1 ·ℎ 𝐵) ∈ ℋ |
11 | 4, 10 | hvaddcli 28590 | . . . . . 6 ⊢ (𝐶 +ℎ (-1 ·ℎ 𝐵)) ∈ ℋ |
12 | 1, 9, 11 | hvassi 28625 | . . . . 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 28625 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) |
14 | 9, 4 | hvcomi 28591 | . . . . . . . . . 10 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
15 | 4, 4 | hvsubvali 28592 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = (𝐶 +ℎ (-1 ·ℎ 𝐶)) |
16 | hvsubid 28598 | . . . . . . . . . . 11 ⊢ (𝐶 ∈ ℋ → (𝐶 −ℎ 𝐶) = 0ℎ) | |
17 | 4, 16 | ax-mp 5 | . . . . . . . . . 10 ⊢ (𝐶 −ℎ 𝐶) = 0ℎ |
18 | 14, 15, 17 | 3eqtr2i 2803 | . . . . . . . . 9 ⊢ ((-1 ·ℎ 𝐶) +ℎ 𝐶) = 0ℎ |
19 | 18 | oveq1i 6985 | . . . . . . . 8 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (0ℎ +ℎ (-1 ·ℎ 𝐵)) |
20 | ax-hv0cl 28575 | . . . . . . . . 9 ⊢ 0ℎ ∈ ℋ | |
21 | 20, 10 | hvcomi 28591 | . . . . . . . 8 ⊢ (0ℎ +ℎ (-1 ·ℎ 𝐵)) = ((-1 ·ℎ 𝐵) +ℎ 0ℎ) |
22 | ax-hvaddid 28576 | . . . . . . . . 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 2801 | . . . . . . 7 ⊢ (((-1 ·ℎ 𝐶) +ℎ 𝐶) +ℎ (-1 ·ℎ 𝐵)) = (-1 ·ℎ 𝐵) |
25 | 13, 24 | eqtr3i 2799 | . . . . . 6 ⊢ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵))) = (-1 ·ℎ 𝐵) |
26 | 25 | oveq2i 6986 | . . . . 5 ⊢ (𝐴 +ℎ ((-1 ·ℎ 𝐶) +ℎ (𝐶 +ℎ (-1 ·ℎ 𝐵)))) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
27 | 7, 12, 26 | 3eqtri 2801 | . . . 4 ⊢ ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) = (𝐴 +ℎ (-1 ·ℎ 𝐵)) |
28 | 3, 27 | eqtr4i 2800 | . . 3 ⊢ (𝐴 −ℎ 𝐵) = ((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵)) |
29 | 28 | fveq2i 6500 | . 2 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) |
30 | 1, 4 | hvsubcli 28593 | . . 3 ⊢ (𝐴 −ℎ 𝐶) ∈ ℋ |
31 | 4, 2 | hvsubcli 28593 | . . 3 ⊢ (𝐶 −ℎ 𝐵) ∈ ℋ |
32 | 30, 31 | norm-ii-i 28709 | . 2 ⊢ (normℎ‘((𝐴 −ℎ 𝐶) +ℎ (𝐶 −ℎ 𝐵))) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
33 | 29, 32 | eqbrtri 4947 | 1 ⊢ (normℎ‘(𝐴 −ℎ 𝐵)) ≤ ((normℎ‘(𝐴 −ℎ 𝐶)) + (normℎ‘(𝐶 −ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∈ wcel 2051 class class class wbr 4926 ‘cfv 6186 (class class class)co 6975 1c1 10335 + caddc 10337 ≤ cle 10474 -cneg 10670 ℋchba 28491 +ℎ cva 28492 ·ℎ csm 28493 normℎcno 28495 0ℎc0v 28496 −ℎ cmv 28497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 ax-pre-sup 10412 ax-hfvadd 28572 ax-hvcom 28573 ax-hvass 28574 ax-hv0cl 28575 ax-hvaddid 28576 ax-hfvmul 28577 ax-hvmulid 28578 ax-hvmulass 28579 ax-hvdistr2 28581 ax-hvmul0 28582 ax-hfi 28651 ax-his1 28654 ax-his2 28655 ax-his3 28656 ax-his4 28657 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-2nd 7501 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-sup 8700 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-nn 11439 df-2 11502 df-3 11503 df-4 11504 df-n0 11707 df-z 11793 df-uz 12058 df-rp 12204 df-seq 13184 df-exp 13244 df-cj 14318 df-re 14319 df-im 14320 df-sqrt 14454 df-abs 14455 df-hnorm 28540 df-hvsub 28543 |
This theorem is referenced by: norm3adifii 28720 norm3lem 28721 norm3dif 28722 |
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