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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 10644 | . . . 4 ⊢ 𝑅 ∈ ℂ |
4 | 1, 3 | mulcli 10637 | . . 3 ⊢ (𝑆 · 𝑅) ∈ ℂ |
5 | normlem1.2 | . . 3 ⊢ 𝐹 ∈ ℋ | |
6 | normlem1.3 | . . 3 ⊢ 𝐺 ∈ ℋ | |
7 | 4, 5, 6 | normlem0 28892 | . 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 14540 | . . . . . . . 8 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · (∗‘𝑅)) |
9 | 3 | cjrebi 14525 | . . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ ↔ (∗‘𝑅) = 𝑅) |
10 | 2, 9 | mpbi 233 | . . . . . . . . 9 ⊢ (∗‘𝑅) = 𝑅 |
11 | 10 | oveq2i 7146 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (∗‘𝑅)) = ((∗‘𝑆) · 𝑅) |
12 | 8, 11 | eqtri 2821 | . . . . . . 7 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · 𝑅) |
13 | 12 | negeqi 10868 | . . . . . 6 ⊢ -(∗‘(𝑆 · 𝑅)) = -((∗‘𝑆) · 𝑅) |
14 | 1 | cjcli 14520 | . . . . . . 7 ⊢ (∗‘𝑆) ∈ ℂ |
15 | 14, 3 | mulneg2i 11076 | . . . . . 6 ⊢ ((∗‘𝑆) · -𝑅) = -((∗‘𝑆) · 𝑅) |
16 | 13, 15 | eqtr4i 2824 | . . . . 5 ⊢ -(∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · -𝑅) |
17 | 16 | oveq1i 7145 | . . . 4 ⊢ (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺)) = (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺)) |
18 | 17 | oveq2i 7146 | . . 3 ⊢ ((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) = ((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) |
19 | 1, 3 | mulneg2i 11076 | . . . . . 6 ⊢ (𝑆 · -𝑅) = -(𝑆 · 𝑅) |
20 | 19 | eqcomi 2807 | . . . . 5 ⊢ -(𝑆 · 𝑅) = (𝑆 · -𝑅) |
21 | 20 | oveq1i 7145 | . . . 4 ⊢ (-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) = ((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) |
22 | 8 | oveq2i 7146 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) |
23 | 3 | cjcli 14520 | . . . . . . . . 9 ⊢ (∗‘𝑅) ∈ ℂ |
24 | 1, 3, 14, 23 | mul4i 10826 | . . . . . . . 8 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) |
25 | normlem1.5 | . . . . . . . . . . . 12 ⊢ (abs‘𝑆) = 1 | |
26 | 25 | oveq1i 7145 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (1↑2) |
27 | 1 | absvalsqi 14745 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (𝑆 · (∗‘𝑆)) |
28 | sq1 13554 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
29 | 26, 27, 28 | 3eqtr3i 2829 | . . . . . . . . . 10 ⊢ (𝑆 · (∗‘𝑆)) = 1 |
30 | 10 | oveq2i 7146 | . . . . . . . . . 10 ⊢ (𝑅 · (∗‘𝑅)) = (𝑅 · 𝑅) |
31 | 29, 30 | oveq12i 7147 | . . . . . . . . 9 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (1 · (𝑅 · 𝑅)) |
32 | 3, 3 | mulcli 10637 | . . . . . . . . . 10 ⊢ (𝑅 · 𝑅) ∈ ℂ |
33 | 32 | mulid2i 10635 | . . . . . . . . 9 ⊢ (1 · (𝑅 · 𝑅)) = (𝑅 · 𝑅) |
34 | 31, 33 | eqtri 2821 | . . . . . . . 8 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (𝑅 · 𝑅) |
35 | 24, 34 | eqtri 2821 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = (𝑅 · 𝑅) |
36 | 22, 35 | eqtri 2821 | . . . . . 6 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅 · 𝑅) |
37 | 3 | sqvali 13539 | . . . . . 6 ⊢ (𝑅↑2) = (𝑅 · 𝑅) |
38 | 36, 37 | eqtr4i 2824 | . . . . 5 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅↑2) |
39 | 38 | oveq1i 7145 | . . . 4 ⊢ (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)) = ((𝑅↑2) · (𝐺 ·ih 𝐺)) |
40 | 21, 39 | oveq12i 7147 | . . 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 7147 | . 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 2821 | 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 1538 ∈ wcel 2111 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 1c1 10527 + caddc 10529 · cmul 10531 -cneg 10860 2c2 11680 ↑cexp 13425 ∗ccj 14447 abscabs 14585 ℋchba 28702 ·ℎ csm 28704 ·ih csp 28705 −ℎ cmv 28708 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-hfvadd 28783 ax-hfvmul 28788 ax-hvmulass 28790 ax-hfi 28862 ax-his1 28865 ax-his2 28866 ax-his3 28867 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-hvsub 28754 |
This theorem is referenced by: normlem4 28896 |
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