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Mirrors > Home > HSE Home > Th. List > norm3adifi | 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, 3-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm3adift.1 | ⊢ 𝐶 ∈ ℋ |
Ref | Expression |
---|---|
norm3adifi | ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvoveq1 7416 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘(𝐴 −ℎ 𝐶)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶))) | |
2 | 1 | fvoveq1d 7415 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) = (abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))))) |
3 | fvoveq1 7416 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → (normℎ‘(𝐴 −ℎ 𝐵)) = (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵))) | |
4 | 2, 3 | breq12d 5154 | . 2 ⊢ (𝐴 = if(𝐴 ∈ ℋ, 𝐴, 0ℎ) → ((abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵)) ↔ (abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)))) |
5 | fvoveq1 7416 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (normℎ‘(𝐵 −ℎ 𝐶)) = (normℎ‘(if(𝐵 ∈ ℋ, 𝐵, 0ℎ) −ℎ 𝐶))) | |
6 | 5 | oveq2d 7409 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶))) = ((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(if(𝐵 ∈ ℋ, 𝐵, 0ℎ) −ℎ 𝐶)))) |
7 | 6 | fveq2d 6882 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → (abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) = (abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(if(𝐵 ∈ ℋ, 𝐵, 0ℎ) −ℎ 𝐶))))) |
8 | oveq2 7401 | . . . 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))) | |
9 | 8 | fveq2d 6882 | . . 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)))) |
10 | 7, 9 | breq12d 5154 | . 2 ⊢ (𝐵 = if(𝐵 ∈ ℋ, 𝐵, 0ℎ) → ((abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐵)) ↔ (abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(if(𝐵 ∈ ℋ, 𝐵, 0ℎ) −ℎ 𝐶)))) ≤ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))))) |
11 | ifhvhv0 30138 | . . 3 ⊢ if(𝐴 ∈ ℋ, 𝐴, 0ℎ) ∈ ℋ | |
12 | ifhvhv0 30138 | . . 3 ⊢ if(𝐵 ∈ ℋ, 𝐵, 0ℎ) ∈ ℋ | |
13 | norm3adift.1 | . . 3 ⊢ 𝐶 ∈ ℋ | |
14 | 11, 12, 13 | norm3adifii 30264 | . 2 ⊢ (abs‘((normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ 𝐶)) − (normℎ‘(if(𝐵 ∈ ℋ, 𝐵, 0ℎ) −ℎ 𝐶)))) ≤ (normℎ‘(if(𝐴 ∈ ℋ, 𝐴, 0ℎ) −ℎ if(𝐵 ∈ ℋ, 𝐵, 0ℎ))) |
15 | 4, 10, 14 | dedth2h 4581 | 1 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (abs‘((normℎ‘(𝐴 −ℎ 𝐶)) − (normℎ‘(𝐵 −ℎ 𝐶)))) ≤ (normℎ‘(𝐴 −ℎ 𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ifcif 4522 class class class wbr 5141 ‘cfv 6532 (class class class)co 7393 ≤ cle 11231 − cmin 11426 abscabs 15163 ℋchba 30035 normℎcno 30039 0ℎc0v 30040 −ℎ cmv 30041 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-hfvadd 30116 ax-hvcom 30117 ax-hvass 30118 ax-hv0cl 30119 ax-hvaddid 30120 ax-hfvmul 30121 ax-hvmulid 30122 ax-hvmulass 30123 ax-hvdistr1 30124 ax-hvdistr2 30125 ax-hvmul0 30126 ax-hfi 30195 ax-his1 30198 ax-his2 30199 ax-his3 30200 ax-his4 30201 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-om 7839 df-2nd 7958 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-er 8686 df-en 8923 df-dom 8924 df-sdom 8925 df-sup 9419 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-n0 12455 df-z 12541 df-uz 12805 df-rp 12957 df-seq 13949 df-exp 14010 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-hnorm 30084 df-hvsub 30087 |
This theorem is referenced by: (None) |
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