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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 2820 | . . . . . 6 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
5 | 1, 2, 3, 4 | normlem2 28886 | . . . . 5 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
6 | 1 | cjcli 14523 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
7 | 2, 3 | hicli 28856 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
8 | 6, 7 | mulcli 10641 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
9 | 3, 2 | hicli 28856 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
10 | 1, 9 | mulcli 10641 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
11 | 8, 10 | addcli 10640 | . . . . . 6 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
12 | 11 | negrebi 10953 | . . . . 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 14734 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
15 | 11 | absnegi 14755 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
16 | 14, 15 | breqtrri 5086 | . 2 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
17 | eqid 2820 | . . 3 ⊢ (𝐺 ·ih 𝐺) = (𝐺 ·ih 𝐺) | |
18 | eqid 2820 | . . 3 ⊢ (𝐹 ·ih 𝐹) = (𝐹 ·ih 𝐹) | |
19 | normlem7.4 | . . 3 ⊢ (abs‘𝑆) = 1 | |
20 | 1, 2, 3, 4, 17, 18, 19 | normlem6 28890 | . 2 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
21 | 11 | negcli 10947 | . . . 4 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
22 | 21 | abscli 14750 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ∈ ℝ |
23 | 2re 11705 | . . . 4 ⊢ 2 ∈ ℝ | |
24 | hiidge0 28873 | . . . . . 6 ⊢ (𝐺 ∈ ℋ → 0 ≤ (𝐺 ·ih 𝐺)) | |
25 | hiidrcl 28870 | . . . . . . . 8 ⊢ (𝐺 ∈ ℋ → (𝐺 ·ih 𝐺) ∈ ℝ) | |
26 | 3, 25 | ax-mp 5 | . . . . . . 7 ⊢ (𝐺 ·ih 𝐺) ∈ ℝ |
27 | 26 | sqrtcli 14726 | . . . . . 6 ⊢ (0 ≤ (𝐺 ·ih 𝐺) → (√‘(𝐺 ·ih 𝐺)) ∈ ℝ) |
28 | 3, 24, 27 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐺 ·ih 𝐺)) ∈ ℝ |
29 | hiidge0 28873 | . . . . . 6 ⊢ (𝐹 ∈ ℋ → 0 ≤ (𝐹 ·ih 𝐹)) | |
30 | hiidrcl 28870 | . . . . . . . 8 ⊢ (𝐹 ∈ ℋ → (𝐹 ·ih 𝐹) ∈ ℝ) | |
31 | 2, 30 | ax-mp 5 | . . . . . . 7 ⊢ (𝐹 ·ih 𝐹) ∈ ℝ |
32 | 31 | sqrtcli 14726 | . . . . . 6 ⊢ (0 ≤ (𝐹 ·ih 𝐹) → (√‘(𝐹 ·ih 𝐹)) ∈ ℝ) |
33 | 2, 29, 32 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐹 ·ih 𝐹)) ∈ ℝ |
34 | 28, 33 | remulcli 10650 | . . . 4 ⊢ ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))) ∈ ℝ |
35 | 23, 34 | remulcli 10650 | . . 3 ⊢ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) ∈ ℝ |
36 | 13, 22, 35 | letri 10762 | . 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 5059 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 ℝcr 10529 0cc0 10530 1c1 10531 + caddc 10533 · cmul 10535 ≤ cle 10669 -cneg 10864 2c2 11686 ∗ccj 14450 √csqrt 14587 abscabs 14588 ℋchba 28694 ·ih csp 28697 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 ax-hfvadd 28775 ax-hv0cl 28778 ax-hfvmul 28780 ax-hvmulass 28782 ax-hvmul0 28785 ax-hfi 28854 ax-his1 28857 ax-his2 28858 ax-his3 28859 ax-his4 28860 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13367 df-exp 13427 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 df-hvsub 28746 |
This theorem is referenced by: normlem7tALT 28894 norm-ii-i 28912 |
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