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Mirrors > Home > HSE Home > Th. List > norm-ii | Structured version Visualization version GIF version |
Description: Triangle inequality for norms. Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 10-Mar-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm-ii | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1 6947 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘(𝐴 +ℎ 𝐵)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵))) | |
2 | fveq2 6448 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘𝐴) = (normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ))) | |
3 | 2 | oveq1d 6939 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((normℎ‘𝐴) + (normℎ‘𝐵)) = ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘𝐵))) |
4 | 1, 3 | breq12d 4901 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵)) ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵)) ≤ ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘𝐵)))) |
5 | oveq2 6932 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵) = (if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) | |
6 | 5 | fveq2d 6452 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))) |
7 | fveq2 6448 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (normℎ‘𝐵) = (normℎ‘if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) | |
8 | 7 | oveq2d 6940 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘𝐵)) = ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘if(𝐵 ∈ ℋ, 𝐵, 0ℎ)))) |
9 | 6, 8 | breq12d 4901 | . 2 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ 𝐵)) ≤ ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘𝐵)) ↔ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ≤ ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘if(𝐵 ∈ ℋ, 𝐵, 0ℎ))))) |
10 | ifhvhv0 28455 | . . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
11 | ifhvhv0 28455 | . . 3 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ | |
12 | 10, 11 | norm-ii-i 28570 | . 2 ⊢ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) +ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) ≤ ((normℎ‘if(𝐴 ∈ ℋ, 𝐴, 0ℎ)) + (normℎ‘if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) |
13 | 4, 9, 12 | dedth2h 4364 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (normℎ‘(𝐴 +ℎ 𝐵)) ≤ ((normℎ‘𝐴) + (normℎ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ifcif 4307 class class class wbr 4888 ‘cfv 6137 (class class class)co 6924 + caddc 10277 ≤ cle 10414 ℋchba 28352 +ℎ cva 28353 normℎcno 28356 0ℎc0v 28357 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 ax-hfvadd 28433 ax-hv0cl 28436 ax-hfvmul 28438 ax-hvmulass 28440 ax-hvmul0 28443 ax-hfi 28512 ax-his1 28515 ax-his2 28516 ax-his3 28517 ax-his4 28518 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-sup 8638 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-n0 11647 df-z 11733 df-uz 11997 df-rp 12142 df-seq 13124 df-exp 13183 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-hnorm 28401 df-hvsub 28404 |
This theorem is referenced by: hhnv 28598 hhssnv 28697 nmoptrii 29529 |
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