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Mirrors > Home > HSE Home > Th. List > norm-iii-i | Structured version Visualization version GIF version |
Description: Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
norm-iii.1 | ⊢ 𝐴 ∈ ℂ |
norm-iii.2 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
norm-iii-i | ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | norm-iii.1 | . . . . 5 ⊢ 𝐴 ∈ ℂ | |
2 | norm-iii.2 | . . . . 5 ⊢ 𝐵 ∈ ℋ | |
3 | 1, 1, 2, 2 | his35i 28865 | . . . 4 ⊢ ((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵)) = ((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵)) |
4 | 3 | fveq2i 6672 | . . 3 ⊢ (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) = (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵))) |
5 | 1 | cjmulrcli 14535 | . . . 4 ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ |
6 | hiidrcl 28871 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 ·ih 𝐵) ∈ ℝ) | |
7 | 2, 6 | ax-mp 5 | . . . 4 ⊢ (𝐵 ·ih 𝐵) ∈ ℝ |
8 | 1 | cjmulge0i 14537 | . . . 4 ⊢ 0 ≤ (𝐴 · (∗‘𝐴)) |
9 | hiidge0 28874 | . . . . 5 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·ih 𝐵)) | |
10 | 2, 9 | ax-mp 5 | . . . 4 ⊢ 0 ≤ (𝐵 ·ih 𝐵) |
11 | 5, 7, 8, 10 | sqrtmulii 14745 | . . 3 ⊢ (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
12 | 4, 11 | eqtri 2844 | . 2 ⊢ (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
13 | 1, 2 | hvmulcli 28790 | . . 3 ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
14 | normval 28900 | . . 3 ⊢ ((𝐴 ·ℎ 𝐵) ∈ ℋ → (normℎ‘(𝐴 ·ℎ 𝐵)) = (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵)))) | |
15 | 13, 14 | ax-mp 5 | . 2 ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) |
16 | absval 14596 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
17 | 1, 16 | ax-mp 5 | . . 3 ⊢ (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))) |
18 | normval 28900 | . . . 4 ⊢ (𝐵 ∈ ℋ → (normℎ‘𝐵) = (√‘(𝐵 ·ih 𝐵))) | |
19 | 2, 18 | ax-mp 5 | . . 3 ⊢ (normℎ‘𝐵) = (√‘(𝐵 ·ih 𝐵)) |
20 | 17, 19 | oveq12i 7167 | . 2 ⊢ ((abs‘𝐴) · (normℎ‘𝐵)) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
21 | 12, 15, 20 | 3eqtr4i 2854 | 1 ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 · cmul 10541 ≤ cle 10675 ∗ccj 14454 √csqrt 14591 abscabs 14592 ℋchba 28695 ·ℎ csm 28697 ·ih csp 28698 normℎcno 28699 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-hv0cl 28779 ax-hfvmul 28781 ax-hvmul0 28786 ax-hfi 28855 ax-his1 28858 ax-his3 28860 ax-his4 28861 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-hnorm 28744 |
This theorem is referenced by: norm-iii 28916 normsubi 28917 normpar2i 28932 |
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