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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 2735 | . . . . . 6 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) = -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) | |
5 | 1, 2, 3, 4 | normlem2 31140 | . . . . 5 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
6 | 1 | cjcli 15205 | . . . . . . . 8 ⊢ (∗‘𝑆) ∈ ℂ |
7 | 2, 3 | hicli 31110 | . . . . . . . 8 ⊢ (𝐹 ·ih 𝐺) ∈ ℂ |
8 | 6, 7 | mulcli 11266 | . . . . . . 7 ⊢ ((∗‘𝑆) · (𝐹 ·ih 𝐺)) ∈ ℂ |
9 | 3, 2 | hicli 31110 | . . . . . . . 8 ⊢ (𝐺 ·ih 𝐹) ∈ ℂ |
10 | 1, 9 | mulcli 11266 | . . . . . . 7 ⊢ (𝑆 · (𝐺 ·ih 𝐹)) ∈ ℂ |
11 | 8, 10 | addcli 11265 | . . . . . 6 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
12 | 11 | negrebi 11581 | . . . . 5 ⊢ (-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ ↔ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ) |
13 | 5, 12 | mpbi 230 | . . . 4 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℝ |
14 | 13 | leabsi 15415 | . . 3 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
15 | 11 | absnegi 15436 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) = (abs‘(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
16 | 14, 15 | breqtrri 5175 | . 2 ⊢ (((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ≤ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) |
17 | eqid 2735 | . . 3 ⊢ (𝐺 ·ih 𝐺) = (𝐺 ·ih 𝐺) | |
18 | eqid 2735 | . . 3 ⊢ (𝐹 ·ih 𝐹) = (𝐹 ·ih 𝐹) | |
19 | normlem7.4 | . . 3 ⊢ (abs‘𝑆) = 1 | |
20 | 1, 2, 3, 4, 17, 18, 19 | normlem6 31144 | . 2 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ≤ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) |
21 | 11 | negcli 11575 | . . . 4 ⊢ -(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹))) ∈ ℂ |
22 | 21 | abscli 15431 | . . 3 ⊢ (abs‘-(((∗‘𝑆) · (𝐹 ·ih 𝐺)) + (𝑆 · (𝐺 ·ih 𝐹)))) ∈ ℝ |
23 | 2re 12338 | . . . 4 ⊢ 2 ∈ ℝ | |
24 | hiidge0 31127 | . . . . . 6 ⊢ (𝐺 ∈ ℋ → 0 ≤ (𝐺 ·ih 𝐺)) | |
25 | hiidrcl 31124 | . . . . . . . 8 ⊢ (𝐺 ∈ ℋ → (𝐺 ·ih 𝐺) ∈ ℝ) | |
26 | 3, 25 | ax-mp 5 | . . . . . . 7 ⊢ (𝐺 ·ih 𝐺) ∈ ℝ |
27 | 26 | sqrtcli 15407 | . . . . . 6 ⊢ (0 ≤ (𝐺 ·ih 𝐺) → (√‘(𝐺 ·ih 𝐺)) ∈ ℝ) |
28 | 3, 24, 27 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐺 ·ih 𝐺)) ∈ ℝ |
29 | hiidge0 31127 | . . . . . 6 ⊢ (𝐹 ∈ ℋ → 0 ≤ (𝐹 ·ih 𝐹)) | |
30 | hiidrcl 31124 | . . . . . . . 8 ⊢ (𝐹 ∈ ℋ → (𝐹 ·ih 𝐹) ∈ ℝ) | |
31 | 2, 30 | ax-mp 5 | . . . . . . 7 ⊢ (𝐹 ·ih 𝐹) ∈ ℝ |
32 | 31 | sqrtcli 15407 | . . . . . 6 ⊢ (0 ≤ (𝐹 ·ih 𝐹) → (√‘(𝐹 ·ih 𝐹)) ∈ ℝ) |
33 | 2, 29, 32 | mp2b 10 | . . . . 5 ⊢ (√‘(𝐹 ·ih 𝐹)) ∈ ℝ |
34 | 28, 33 | remulcli 11275 | . . . 4 ⊢ ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹))) ∈ ℝ |
35 | 23, 34 | remulcli 11275 | . . 3 ⊢ (2 · ((√‘(𝐺 ·ih 𝐺)) · (√‘(𝐹 ·ih 𝐹)))) ∈ ℝ |
36 | 13, 22, 35 | letri 11388 | . 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 692 | 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 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 · cmul 11158 ≤ cle 11294 -cneg 11491 2c2 12319 ∗ccj 15132 √csqrt 15269 abscabs 15270 ℋchba 30948 ·ih csp 30951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-hfvadd 31029 ax-hv0cl 31032 ax-hfvmul 31034 ax-hvmulass 31036 ax-hvmul0 31039 ax-hfi 31108 ax-his1 31111 ax-his2 31112 ax-his3 31113 ax-his4 31114 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-hvsub 31000 |
This theorem is referenced by: normlem7tALT 31148 norm-ii-i 31166 |
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