Hilbert Space Explorer |
< Previous
Next >
Nearby theorems |
||
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 2824 | . . . . . 6 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
5 | 1, 2, 3, 4 | normlem2 28891 | . . . . 5 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
6 | 1 | cjcli 14531 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
7 | 2, 3 | hicli 28861 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
8 | 6, 7 | mulcli 10651 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
9 | 3, 2 | hicli 28861 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
10 | 1, 9 | mulcli 10651 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
11 | 8, 10 | addcli 10650 | . . . . . 6 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
12 | 11 | negrebi 10963 | . . . . 5 ⊢ (-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ ↔ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ) |
13 | 5, 12 | mpbi 232 | . . . 4 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
14 | 13 | leabsi 14742 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
15 | 11 | absnegi 14763 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
16 | 14, 15 | breqtrri 5096 | . 2 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
17 | eqid 2824 | . . 3 ⊢ (𝐺 ·ih 𝐺) = (𝐺 ·ih 𝐺) | |
18 | eqid 2824 | . . 3 ⊢ (𝐹 ·ih 𝐹) = (𝐹 ·ih 𝐹) | |
19 | normlem7.4 | . . 3 ⊢ (abs‘𝑆) = 1 | |
20 | 1, 2, 3, 4, 17, 18, 19 | normlem6 28895 | . 2 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
21 | 11 | negcli 10957 | . . . 4 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
22 | 21 | abscli 14758 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ∈ ℝ |
23 | 2re 11714 | . . . 4 ⊢ 2 ∈ ℝ | |
24 | hiidge0 28878 | . . . . . 6 ⊢ (𝐺 ∈ ℋ → 0 ≤ (𝐺 ·ih 𝐺)) | |
25 | hiidrcl 28875 | . . . . . . . 8 ⊢ (𝐺 ∈ ℋ → (𝐺 ·ih 𝐺) ∈ ℝ) | |
26 | 3, 25 | ax-mp 5 | . . . . . . 7 ⊢ (𝐺 ·ih 𝐺) ∈ ℝ |
27 | 26 | sqrtcli 14734 | . . . . . 6 ⊢ (0 ≤ (𝐺 ·ih 𝐺) → (√‘(𝐺 ·ih 𝐺)) ∈ ℝ) |
28 | 3, 24, 27 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐺 ·ih 𝐺)) ∈ ℝ |
29 | hiidge0 28878 | . . . . . 6 ⊢ (𝐹 ∈ ℋ → 0 ≤ (𝐹 ·ih 𝐹)) | |
30 | hiidrcl 28875 | . . . . . . . 8 ⊢ (𝐹 ∈ ℋ → (𝐹 ·ih 𝐹) ∈ ℝ) | |
31 | 2, 30 | ax-mp 5 | . . . . . . 7 ⊢ (𝐹 ·ih 𝐹) ∈ ℝ |
32 | 31 | sqrtcli 14734 | . . . . . 6 ⊢ (0 ≤ (𝐹 ·ih 𝐹) → (√‘(𝐹 ·ih 𝐹)) ∈ ℝ) |
33 | 2, 29, 32 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐹 ·ih 𝐹)) ∈ ℝ |
34 | 28, 33 | remulcli 10660 | . . . 4 ⊢ ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))) ∈ ℝ |
35 | 23, 34 | remulcli 10660 | . . 3 ⊢ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) ∈ ℝ |
36 | 13, 22, 35 | letri 10772 | . 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 690 | 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 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 1c1 10541 + caddc 10543 · cmul 10545 ≤ cle 10679 -cneg 10874 2c2 11695 ∗ccj 14458 √csqrt 14595 abscabs 14596 ℋchba 28699 ·ih csp 28702 |
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-hv0cl 28783 ax-hfvmul 28785 ax-hvmulass 28787 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-hvsub 28751 |
This theorem is referenced by: normlem7tALT 28899 norm-ii-i 28917 |
Copyright terms: Public domain | W3C validator |