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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 11278 | . . . . . 6 ⊢ 𝑅 ∈ ℂ |
5 | 2, 4 | mulcli 11271 | . . . . 5 ⊢ (𝑆 · 𝑅) ∈ ℂ |
6 | normlem1.3 | . . . . 5 ⊢ 𝐺 ∈ ℋ | |
7 | 5, 6 | hvmulcli 30947 | . . . 4 ⊢ ((𝑆 · 𝑅) ·ℎ 𝐺) ∈ ℋ |
8 | 1, 7 | hvsubcli 30954 | . . 3 ⊢ (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ∈ ℋ |
9 | hiidge0 31031 | . . 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 31046 | . 2 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) |
16 | 10, 15 | breqtri 5178 | 1 ⊢ 0 ≤ (((𝐴 · (𝑅↑2)) + (𝐵 · 𝑅)) + 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 ≤ cle 11299 -cneg 11495 2c2 12319 ↑cexp 14081 ∗ccj 15101 abscabs 15239 ℋchba 30852 ·ℎ csm 30854 ·ih csp 30855 −ℎ cmv 30858 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-hfvadd 30933 ax-hv0cl 30936 ax-hfvmul 30938 ax-hvmulass 30940 ax-hvmul0 30943 ax-hfi 31012 ax-his1 31015 ax-his2 31016 ax-his3 31017 ax-his4 31018 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-sup 9485 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12611 df-uz 12875 df-rp 13029 df-seq 14022 df-exp 14082 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-hvsub 30904 |
This theorem is referenced by: normlem6 31048 |
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