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Mirrors > Home > HSE Home > Th. List > normlem1 | 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, 22-Aug-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
normlem1.1 | ⊢ 𝑆 ∈ ℂ |
normlem1.2 | ⊢ 𝐹 ∈ ℋ |
normlem1.3 | ⊢ 𝐺 ∈ ℋ |
normlem1.4 | ⊢ 𝑅 ∈ ℝ |
normlem1.5 | ⊢ (abs‘𝑆) = 1 |
Ref | Expression |
---|---|
normlem1 | ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | normlem1.1 | . . . 4 ⊢ 𝑆 ∈ ℂ | |
2 | normlem1.4 | . . . . 5 ⊢ 𝑅 ∈ ℝ | |
3 | 2 | recni 11268 | . . . 4 ⊢ 𝑅 ∈ ℂ |
4 | 1, 3 | mulcli 11261 | . . 3 ⊢ (𝑆 · 𝑅) ∈ ℂ |
5 | normlem1.2 | . . 3 ⊢ 𝐹 ∈ ℋ | |
6 | normlem1.3 | . . 3 ⊢ 𝐺 ∈ ℋ | |
7 | 4, 5, 6 | normlem0 31038 | . 2 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) + ((-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) + (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)))) |
8 | 1, 3 | cjmuli 15188 | . . . . . . . 8 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · (∗‘𝑅)) |
9 | 3 | cjrebi 15173 | . . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ ↔ (∗‘𝑅) = 𝑅) |
10 | 2, 9 | mpbi 229 | . . . . . . . . 9 ⊢ (∗‘𝑅) = 𝑅 |
11 | 10 | oveq2i 7426 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (∗‘𝑅)) = ((∗‘𝑆) · 𝑅) |
12 | 8, 11 | eqtri 2754 | . . . . . . 7 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · 𝑅) |
13 | 12 | negeqi 11493 | . . . . . 6 ⊢ -(∗‘(𝑆 · 𝑅)) = -((∗‘𝑆) · 𝑅) |
14 | 1 | cjcli 15168 | . . . . . . 7 ⊢ (∗‘𝑆) ∈ ℂ |
15 | 14, 3 | mulneg2i 11701 | . . . . . 6 ⊢ ((∗‘𝑆) · -𝑅) = -((∗‘𝑆) · 𝑅) |
16 | 13, 15 | eqtr4i 2757 | . . . . 5 ⊢ -(∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · -𝑅) |
17 | 16 | oveq1i 7425 | . . . 4 ⊢ (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺)) = (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺)) |
18 | 17 | oveq2i 7426 | . . 3 ⊢ ((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) = ((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) |
19 | 1, 3 | mulneg2i 11701 | . . . . . 6 ⊢ (𝑆 · -𝑅) = -(𝑆 · 𝑅) |
20 | 19 | eqcomi 2735 | . . . . 5 ⊢ -(𝑆 · 𝑅) = (𝑆 · -𝑅) |
21 | 20 | oveq1i 7425 | . . . 4 ⊢ (-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) = ((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) |
22 | 8 | oveq2i 7426 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) |
23 | 3 | cjcli 15168 | . . . . . . . . 9 ⊢ (∗‘𝑅) ∈ ℂ |
24 | 1, 3, 14, 23 | mul4i 11451 | . . . . . . . 8 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) |
25 | normlem1.5 | . . . . . . . . . . . 12 ⊢ (abs‘𝑆) = 1 | |
26 | 25 | oveq1i 7425 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (1↑2) |
27 | 1 | absvalsqi 15392 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (𝑆 · (∗‘𝑆)) |
28 | sq1 14206 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
29 | 26, 27, 28 | 3eqtr3i 2762 | . . . . . . . . . 10 ⊢ (𝑆 · (∗‘𝑆)) = 1 |
30 | 10 | oveq2i 7426 | . . . . . . . . . 10 ⊢ (𝑅 · (∗‘𝑅)) = (𝑅 · 𝑅) |
31 | 29, 30 | oveq12i 7427 | . . . . . . . . 9 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (1 · (𝑅 · 𝑅)) |
32 | 3, 3 | mulcli 11261 | . . . . . . . . . 10 ⊢ (𝑅 · 𝑅) ∈ ℂ |
33 | 32 | mullidi 11259 | . . . . . . . . 9 ⊢ (1 · (𝑅 · 𝑅)) = (𝑅 · 𝑅) |
34 | 31, 33 | eqtri 2754 | . . . . . . . 8 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (𝑅 · 𝑅) |
35 | 24, 34 | eqtri 2754 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = (𝑅 · 𝑅) |
36 | 22, 35 | eqtri 2754 | . . . . . 6 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅 · 𝑅) |
37 | 3 | sqvali 14191 | . . . . . 6 ⊢ (𝑅↑2) = (𝑅 · 𝑅) |
38 | 36, 37 | eqtr4i 2757 | . . . . 5 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅↑2) |
39 | 38 | oveq1i 7425 | . . . 4 ⊢ (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)) = ((𝑅↑2) · (𝐺 ·ih 𝐺)) |
40 | 21, 39 | oveq12i 7427 | . . 3 ⊢ ((-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) + (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺))) = (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺))) |
41 | 18, 40 | oveq12i 7427 | . 2 ⊢ (((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) + ((-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) + (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) |
42 | 7, 41 | eqtri 2754 | 1 ⊢ ((𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺)) ·ih (𝐹 −ℎ ((𝑆 · 𝑅) ·ℎ 𝐺))) = (((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) + (((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) + ((𝑅↑2) · (𝐺 ·ih 𝐺)))) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ∈ wcel 2099 ‘cfv 6545 (class class class)co 7415 ℂcc 11146 ℝcr 11147 1c1 11149 + caddc 11151 · cmul 11153 -cneg 11485 2c2 12312 ↑cexp 14074 ∗ccj 15095 abscabs 15233 ℋchba 30848 ·ℎ csm 30850 ·ih csp 30851 −ℎ cmv 30854 |
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 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7737 ax-cnex 11204 ax-resscn 11205 ax-1cn 11206 ax-icn 11207 ax-addcl 11208 ax-addrcl 11209 ax-mulcl 11210 ax-mulrcl 11211 ax-mulcom 11212 ax-addass 11213 ax-mulass 11214 ax-distr 11215 ax-i2m1 11216 ax-1ne0 11217 ax-1rid 11218 ax-rnegex 11219 ax-rrecex 11220 ax-cnre 11221 ax-pre-lttri 11222 ax-pre-lttrn 11223 ax-pre-ltadd 11224 ax-pre-mulgt0 11225 ax-pre-sup 11226 ax-hfvadd 30929 ax-hfvmul 30934 ax-hvmulass 30936 ax-hfi 31008 ax-his1 31011 ax-his2 31012 ax-his3 31013 |
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 3365 df-reu 3366 df-rab 3421 df-v 3466 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3968 df-nul 4325 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4908 df-iun 4997 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6497 df-fun 6547 df-fn 6548 df-f 6549 df-f1 6550 df-fo 6551 df-f1o 6552 df-fv 6553 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8966 df-dom 8967 df-sdom 8968 df-sup 9477 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-sub 11486 df-neg 11487 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-n0 12518 df-z 12604 df-uz 12868 df-rp 13022 df-seq 14015 df-exp 14075 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-hvsub 30900 |
This theorem is referenced by: normlem4 31042 |
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