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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 10643 | . . . 4 ⊢ 𝑅 ∈ ℂ |
4 | 1, 3 | mulcli 10636 | . . 3 ⊢ (𝑆 · 𝑅) ∈ ℂ |
5 | normlem1.2 | . . 3 ⊢ 𝐹 ∈ ℋ | |
6 | normlem1.3 | . . 3 ⊢ 𝐺 ∈ ℋ | |
7 | 4, 5, 6 | normlem0 28813 | . 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 14536 | . . . . . . . 8 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · (∗‘𝑅)) |
9 | 3 | cjrebi 14521 | . . . . . . . . . 10 ⊢ (𝑅 ∈ ℝ ↔ (∗‘𝑅) = 𝑅) |
10 | 2, 9 | mpbi 231 | . . . . . . . . 9 ⊢ (∗‘𝑅) = 𝑅 |
11 | 10 | oveq2i 7156 | . . . . . . . 8 ⊢ ((∗‘𝑆) · (∗‘𝑅)) = ((∗‘𝑆) · 𝑅) |
12 | 8, 11 | eqtri 2841 | . . . . . . 7 ⊢ (∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · 𝑅) |
13 | 12 | negeqi 10867 | . . . . . 6 ⊢ -(∗‘(𝑆 · 𝑅)) = -((∗‘𝑆) · 𝑅) |
14 | 1 | cjcli 14516 | . . . . . . 7 ⊢ (∗‘𝑆) ∈ ℂ |
15 | 14, 3 | mulneg2i 11075 | . . . . . 6 ⊢ ((∗‘𝑆) · -𝑅) = -((∗‘𝑆) · 𝑅) |
16 | 13, 15 | eqtr4i 2844 | . . . . 5 ⊢ -(∗‘(𝑆 · 𝑅)) = ((∗‘𝑆) · -𝑅) |
17 | 16 | oveq1i 7155 | . . . 4 ⊢ (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺)) = (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺)) |
18 | 17 | oveq2i 7156 | . . 3 ⊢ ((𝐹 ·ih 𝐹) + (-(∗‘(𝑆 · 𝑅)) · (𝐹 ·ih 𝐺))) = ((𝐹 ·ih 𝐹) + (((∗‘𝑆) · -𝑅) · (𝐹 ·ih 𝐺))) |
19 | 1, 3 | mulneg2i 11075 | . . . . . 6 ⊢ (𝑆 · -𝑅) = -(𝑆 · 𝑅) |
20 | 19 | eqcomi 2827 | . . . . 5 ⊢ -(𝑆 · 𝑅) = (𝑆 · -𝑅) |
21 | 20 | oveq1i 7155 | . . . 4 ⊢ (-(𝑆 · 𝑅) · (𝐺 ·ih 𝐹)) = ((𝑆 · -𝑅) · (𝐺 ·ih 𝐹)) |
22 | 8 | oveq2i 7156 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) |
23 | 3 | cjcli 14516 | . . . . . . . . 9 ⊢ (∗‘𝑅) ∈ ℂ |
24 | 1, 3, 14, 23 | mul4i 10825 | . . . . . . . 8 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) |
25 | normlem1.5 | . . . . . . . . . . . 12 ⊢ (abs‘𝑆) = 1 | |
26 | 25 | oveq1i 7155 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (1↑2) |
27 | 1 | absvalsqi 14741 | . . . . . . . . . . 11 ⊢ ((abs‘𝑆)↑2) = (𝑆 · (∗‘𝑆)) |
28 | sq1 13546 | . . . . . . . . . . 11 ⊢ (1↑2) = 1 | |
29 | 26, 27, 28 | 3eqtr3i 2849 | . . . . . . . . . 10 ⊢ (𝑆 · (∗‘𝑆)) = 1 |
30 | 10 | oveq2i 7156 | . . . . . . . . . 10 ⊢ (𝑅 · (∗‘𝑅)) = (𝑅 · 𝑅) |
31 | 29, 30 | oveq12i 7157 | . . . . . . . . 9 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (1 · (𝑅 · 𝑅)) |
32 | 3, 3 | mulcli 10636 | . . . . . . . . . 10 ⊢ (𝑅 · 𝑅) ∈ ℂ |
33 | 32 | mulid2i 10634 | . . . . . . . . 9 ⊢ (1 · (𝑅 · 𝑅)) = (𝑅 · 𝑅) |
34 | 31, 33 | eqtri 2841 | . . . . . . . 8 ⊢ ((𝑆 · (∗‘𝑆)) · (𝑅 · (∗‘𝑅))) = (𝑅 · 𝑅) |
35 | 24, 34 | eqtri 2841 | . . . . . . 7 ⊢ ((𝑆 · 𝑅) · ((∗‘𝑆) · (∗‘𝑅))) = (𝑅 · 𝑅) |
36 | 22, 35 | eqtri 2841 | . . . . . 6 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅 · 𝑅) |
37 | 3 | sqvali 13531 | . . . . . 6 ⊢ (𝑅↑2) = (𝑅 · 𝑅) |
38 | 36, 37 | eqtr4i 2844 | . . . . 5 ⊢ ((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) = (𝑅↑2) |
39 | 38 | oveq1i 7155 | . . . 4 ⊢ (((𝑆 · 𝑅) · (∗‘(𝑆 · 𝑅))) · (𝐺 ·ih 𝐺)) = ((𝑅↑2) · (𝐺 ·ih 𝐺)) |
40 | 21, 39 | oveq12i 7157 | . . 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 7157 | . 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 2841 | 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 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 1c1 10526 + caddc 10528 · cmul 10530 -cneg 10859 2c2 11680 ↑cexp 13417 ∗ccj 14443 abscabs 14581 ℋchba 28623 ·ℎ csm 28625 ·ih csp 28626 −ℎ cmv 28629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-hfvadd 28704 ax-hfvmul 28709 ax-hvmulass 28711 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-hvsub 28675 |
This theorem is referenced by: normlem4 28817 |
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