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Mirrors > Home > HSE Home > Th. List > normlem5 | 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, 10-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normlem1.1 | ⊢ 𝑆 ∈ ℂ |
normlem1.2 | ⊢ 𝐹 ∈ ℋ |
normlem1.3 | ⊢ 𝐺 ∈ ℋ |
normlem2.4 | ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) |
normlem3.5 | ⊢ 𝐴 = (𝐺 ·ih 𝐺) |
normlem3.6 | ⊢ 𝐶 = (𝐹 ·ih 𝐹) |
normlem4.7 | ⊢ 𝑅 ∈ ℝ |
normlem4.8 | ⊢ (abs‘𝑆) = 1 |
Ref | Expression |
---|---|
normlem5 | ⊢ 0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normlem1.2 | . . . 4 ⊢ 𝐹 ∈ ℋ | |
2 | normlem1.1 | . . . . . 6 ⊢ 𝑆 ∈ ℂ | |
3 | normlem4.7 | . . . . . . 7 ⊢ 𝑅 ∈ ℝ | |
4 | 3 | recni 11169 | . . . . . 6 ⊢ 𝑅 ∈ ℂ |
5 | 2, 4 | mulcli 11162 | . . . . 5 ⊢ (𝑆 · 𝑅) ∈ ℂ |
6 | normlem1.3 | . . . . 5 ⊢ 𝐺 ∈ ℋ | |
7 | 5, 6 | hvmulcli 29956 | . . . 4 ⊢ ((𝑆 · 𝑅) ·ℎ 𝐺) ∈ ℋ |
8 | 1, 7 | hvsubcli 29963 | . . 3 ⊢ (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ∈ ℋ |
9 | hiidge0 30040 | . . 3 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ∈ ℋ → 0 ≤ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)))) | |
10 | 8, 9 | ax-mp 5 | . 2 ⊢ 0 ≤ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) |
11 | normlem2.4 | . . 3 ⊢ 𝐵 = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
12 | normlem3.5 | . . 3 ⊢ 𝐴 = (𝐺 ·ih 𝐺) | |
13 | normlem3.6 | . . 3 ⊢ 𝐶 = (𝐹 ·ih 𝐹) | |
14 | normlem4.8 | . . 3 ⊢ (abs‘𝑆) = 1 | |
15 | 2, 1, 6, 11, 12, 13, 3, 14 | normlem4 30055 | . 2 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) |
16 | 10, 15 | breqtri 5130 | 1 ⊢ 0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 · cmul 11056 ≤ cle 11190 -cneg 11386 2c2 12208 ↑cexp 13967 ∗ccj 14981 abscabs 15119 ℋchba 29861 ·ℎ csm 29863 ·ih csp 29864 −ℎ cmv 29867 |
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 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-hfvadd 29942 ax-hv0cl 29945 ax-hfvmul 29947 ax-hvmulass 29949 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-hvsub 29913 |
This theorem is referenced by: normlem6 30057 |
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