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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 2622 | . . . . . 6 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
5 | 1, 2, 3, 4 | normlem2 27968 | . . . . 5 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
6 | 1 | cjcli 13909 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
7 | 2, 3 | hicli 27938 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
8 | 6, 7 | mulcli 10045 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
9 | 3, 2 | hicli 27938 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
10 | 1, 9 | mulcli 10045 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
11 | 8, 10 | addcli 10044 | . . . . . 6 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
12 | 11 | negrebi 10355 | . . . . 5 ⊢ (-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ ↔ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ) |
13 | 5, 12 | mpbi 220 | . . . 4 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
14 | 13 | leabsi 14119 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
15 | 11 | absnegi 14139 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
16 | 14, 15 | breqtrri 4680 | . 2 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
17 | eqid 2622 | . . 3 ⊢ (𝐺 ·ih 𝐺) = (𝐺 ·ih 𝐺) | |
18 | eqid 2622 | . . 3 ⊢ (𝐹 ·ih 𝐹) = (𝐹 ·ih 𝐹) | |
19 | normlem7.4 | . . 3 ⊢ (abs‘𝑆) = 1 | |
20 | 1, 2, 3, 4, 17, 18, 19 | normlem6 27972 | . 2 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
21 | 11 | negcli 10349 | . . . 4 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
22 | 21 | abscli 14134 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ∈ ℝ |
23 | 2re 11090 | . . . 4 ⊢ 2 ∈ ℝ | |
24 | hiidge0 27955 | . . . . . 6 ⊢ (𝐺 ∈ ℋ → 0 ≤ (𝐺 ·ih 𝐺)) | |
25 | hiidrcl 27952 | . . . . . . . 8 ⊢ (𝐺 ∈ ℋ → (𝐺 ·ih 𝐺) ∈ ℝ) | |
26 | 3, 25 | ax-mp 5 | . . . . . . 7 ⊢ (𝐺 ·ih 𝐺) ∈ ℝ |
27 | 26 | sqrtcli 14111 | . . . . . 6 ⊢ (0 ≤ (𝐺 ·ih 𝐺) → (√‘(𝐺 ·ih 𝐺)) ∈ ℝ) |
28 | 3, 24, 27 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐺 ·ih 𝐺)) ∈ ℝ |
29 | hiidge0 27955 | . . . . . 6 ⊢ (𝐹 ∈ ℋ → 0 ≤ (𝐹 ·ih 𝐹)) | |
30 | hiidrcl 27952 | . . . . . . . 8 ⊢ (𝐹 ∈ ℋ → (𝐹 ·ih 𝐹) ∈ ℝ) | |
31 | 2, 30 | ax-mp 5 | . . . . . . 7 ⊢ (𝐹 ·ih 𝐹) ∈ ℝ |
32 | 31 | sqrtcli 14111 | . . . . . 6 ⊢ (0 ≤ (𝐹 ·ih 𝐹) → (√‘(𝐹 ·ih 𝐹)) ∈ ℝ) |
33 | 2, 29, 32 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐹 ·ih 𝐹)) ∈ ℝ |
34 | 28, 33 | remulcli 10054 | . . . 4 ⊢ ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))) ∈ ℝ |
35 | 23, 34 | remulcli 10054 | . . 3 ⊢ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) ∈ ℝ |
36 | 13, 22, 35 | letri 10166 | . 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 708 | 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 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ≤ cle 10075 -cneg 10267 2c2 11070 ∗ccj 13836 √csqrt 13973 abscabs 13974 ℋchil 27776 ·ih csp 27779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmulass 27864 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his2 27940 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hvsub 27828 |
This theorem is referenced by: normlem7tALT 27976 norm-ii-i 27994 |
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