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Mirrors > Home > HSE Home > Th. List > normlem7 | Structured version Visualization version GIF version |
Description: Lemma used to derive properties of norm. Part of Theorem 3.3(ii) of [Beran] p. 97. (Contributed by NM, 11-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normlem1.1 | ⊢ 𝑆 ∈ ℂ |
normlem1.2 | ⊢ 𝐹 ∈ ℋ |
normlem1.3 | ⊢ 𝐺 ∈ ℋ |
normlem7.4 | ⊢ (abs‘𝑆) = 1 |
Ref | Expression |
---|---|
normlem7 | ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normlem1.1 | . . . . . 6 ⊢ 𝑆 ∈ ℂ | |
2 | normlem1.2 | . . . . . 6 ⊢ 𝐹 ∈ ℋ | |
3 | normlem1.3 | . . . . . 6 ⊢ 𝐺 ∈ ℋ | |
4 | eqid 2798 | . . . . . 6 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
5 | 1, 2, 3, 4 | normlem2 28894 | . . . . 5 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
6 | 1 | cjcli 14520 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
7 | 2, 3 | hicli 28864 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
8 | 6, 7 | mulcli 10637 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
9 | 3, 2 | hicli 28864 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
10 | 1, 9 | mulcli 10637 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
11 | 8, 10 | addcli 10636 | . . . . . 6 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
12 | 11 | negrebi 10949 | . . . . 5 ⊢ (-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ ↔ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ) |
13 | 5, 12 | mpbi 233 | . . . 4 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
14 | 13 | leabsi 14731 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
15 | 11 | absnegi 14752 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
16 | 14, 15 | breqtrri 5057 | . 2 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
17 | eqid 2798 | . . 3 ⊢ (𝐺 ·ih 𝐺) = (𝐺 ·ih 𝐺) | |
18 | eqid 2798 | . . 3 ⊢ (𝐹 ·ih 𝐹) = (𝐹 ·ih 𝐹) | |
19 | normlem7.4 | . . 3 ⊢ (abs‘𝑆) = 1 | |
20 | 1, 2, 3, 4, 17, 18, 19 | normlem6 28898 | . 2 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
21 | 11 | negcli 10943 | . . . 4 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
22 | 21 | abscli 14747 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ∈ ℝ |
23 | 2re 11699 | . . . 4 ⊢ 2 ∈ ℝ | |
24 | hiidge0 28881 | . . . . . 6 ⊢ (𝐺 ∈ ℋ → 0 ≤ (𝐺 ·ih 𝐺)) | |
25 | hiidrcl 28878 | . . . . . . . 8 ⊢ (𝐺 ∈ ℋ → (𝐺 ·ih 𝐺) ∈ ℝ) | |
26 | 3, 25 | ax-mp 5 | . . . . . . 7 ⊢ (𝐺 ·ih 𝐺) ∈ ℝ |
27 | 26 | sqrtcli 14723 | . . . . . 6 ⊢ (0 ≤ (𝐺 ·ih 𝐺) → (√‘(𝐺 ·ih 𝐺)) ∈ ℝ) |
28 | 3, 24, 27 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐺 ·ih 𝐺)) ∈ ℝ |
29 | hiidge0 28881 | . . . . . 6 ⊢ (𝐹 ∈ ℋ → 0 ≤ (𝐹 ·ih 𝐹)) | |
30 | hiidrcl 28878 | . . . . . . . 8 ⊢ (𝐹 ∈ ℋ → (𝐹 ·ih 𝐹) ∈ ℝ) | |
31 | 2, 30 | ax-mp 5 | . . . . . . 7 ⊢ (𝐹 ·ih 𝐹) ∈ ℝ |
32 | 31 | sqrtcli 14723 | . . . . . 6 ⊢ (0 ≤ (𝐹 ·ih 𝐹) → (√‘(𝐹 ·ih 𝐹)) ∈ ℝ) |
33 | 2, 29, 32 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐹 ·ih 𝐹)) ∈ ℝ |
34 | 28, 33 | remulcli 10646 | . . . 4 ⊢ ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))) ∈ ℝ |
35 | 23, 34 | remulcli 10646 | . . 3 ⊢ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) ∈ ℝ |
36 | 13, 22, 35 | letri 10758 | . 2 ⊢ (((((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ∧ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))))) → (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))))) |
37 | 16, 20, 36 | mp2an 691 | 1 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 -cneg 10860 2c2 11680 ∗ccj 14447 √csqrt 14584 abscabs 14585 ℋchba 28702 ·ih csp 28705 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-hfvadd 28783 ax-hv0cl 28786 ax-hfvmul 28788 ax-hvmulass 28790 ax-hvmul0 28793 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 ax-his4 28868 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-hvsub 28754 |
This theorem is referenced by: normlem7tALT 28902 norm-ii-i 28920 |
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