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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 27946 | . . . 4 ⊢ ((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵)) = ((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵)) |
4 | 3 | fveq2i 6194 | . . 3 ⊢ (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) = (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵))) |
5 | 1 | cjmulrcli 13917 | . . . 4 ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ |
6 | hiidrcl 27952 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 ·ih 𝐵) ∈ ℝ) | |
7 | 2, 6 | ax-mp 5 | . . . 4 ⊢ (𝐵 ·ih 𝐵) ∈ ℝ |
8 | 1 | cjmulge0i 13919 | . . . 4 ⊢ 0 ≤ (𝐴 · (∗‘𝐴)) |
9 | hiidge0 27955 | . . . . 5 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·ih 𝐵)) | |
10 | 2, 9 | ax-mp 5 | . . . 4 ⊢ 0 ≤ (𝐵 ·ih 𝐵) |
11 | 5, 7, 8, 10 | sqrtmulii 14126 | . . 3 ⊢ (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
12 | 4, 11 | eqtri 2644 | . 2 ⊢ (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
13 | 1, 2 | hvmulcli 27871 | . . 3 ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
14 | normval 27981 | . . 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 13978 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
17 | 1, 16 | ax-mp 5 | . . 3 ⊢ (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))) |
18 | normval 27981 | . . . 4 ⊢ (𝐵 ∈ ℋ → (normℎ‘𝐵) = (√‘(𝐵 ·ih 𝐵))) | |
19 | 2, 18 | ax-mp 5 | . . 3 ⊢ (normℎ‘𝐵) = (√‘(𝐵 ·ih 𝐵)) |
20 | 17, 19 | oveq12i 6662 | . 2 ⊢ ((abs‘𝐴) · (normℎ‘𝐵)) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
21 | 12, 15, 20 | 3eqtr4i 2654 | 1 ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 ≤ cle 10075 ∗ccj 13836 √csqrt 13973 abscabs 13974 ℋchil 27776 ·ℎ csm 27778 ·ih csp 27779 normℎcno 27780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmul0 27867 ax-hfi 27936 ax-his1 27939 ax-his3 27941 ax-his4 27942 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-hnorm 27825 |
This theorem is referenced by: norm-iii 27997 normsubi 27998 normpar2i 28013 |
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