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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 28255 | . . . 4 ⊢ ((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵)) = ((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵)) |
4 | 3 | fveq2i 6355 | . . 3 ⊢ (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) = (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵))) |
5 | 1 | cjmulrcli 14116 | . . . 4 ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ |
6 | hiidrcl 28261 | . . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 ·ih 𝐵) ∈ ℝ) | |
7 | 2, 6 | ax-mp 5 | . . . 4 ⊢ (𝐵 ·ih 𝐵) ∈ ℝ |
8 | 1 | cjmulge0i 14118 | . . . 4 ⊢ 0 ≤ (𝐴 · (∗‘𝐴)) |
9 | hiidge0 28264 | . . . . 5 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·ih 𝐵)) | |
10 | 2, 9 | ax-mp 5 | . . . 4 ⊢ 0 ≤ (𝐵 ·ih 𝐵) |
11 | 5, 7, 8, 10 | sqrtmulii 14325 | . . 3 ⊢ (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·ih 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
12 | 4, 11 | eqtri 2782 | . 2 ⊢ (√‘((𝐴 ·ℎ 𝐵) ·ih (𝐴 ·ℎ 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
13 | 1, 2 | hvmulcli 28180 | . . 3 ⊢ (𝐴 ·ℎ 𝐵) ∈ ℋ |
14 | normval 28290 | . . 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 14177 | . . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
17 | 1, 16 | ax-mp 5 | . . 3 ⊢ (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))) |
18 | normval 28290 | . . . 4 ⊢ (𝐵 ∈ ℋ → (normℎ‘𝐵) = (√‘(𝐵 ·ih 𝐵))) | |
19 | 2, 18 | ax-mp 5 | . . 3 ⊢ (normℎ‘𝐵) = (√‘(𝐵 ·ih 𝐵)) |
20 | 17, 19 | oveq12i 6825 | . 2 ⊢ ((abs‘𝐴) · (normℎ‘𝐵)) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·ih 𝐵))) |
21 | 12, 15, 20 | 3eqtr4i 2792 | 1 ⊢ (normℎ‘(𝐴 ·ℎ 𝐵)) = ((abs‘𝐴) · (normℎ‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∈ wcel 2139 class class class wbr 4804 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 ℝcr 10127 0cc0 10128 · cmul 10133 ≤ cle 10267 ∗ccj 14035 √csqrt 14172 abscabs 14173 ℋchil 28085 ·ℎ csm 28087 ·ih csp 28088 normℎcno 28089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-hv0cl 28169 ax-hfvmul 28171 ax-hvmul0 28176 ax-hfi 28245 ax-his1 28248 ax-his3 28250 ax-his4 28251 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-2nd 7334 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-er 7911 df-en 8122 df-dom 8123 df-sdom 8124 df-sup 8513 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-n0 11485 df-z 11570 df-uz 11880 df-rp 12026 df-seq 12996 df-exp 13055 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-hnorm 28134 |
This theorem is referenced by: norm-iii 28306 normsubi 28307 normpar2i 28322 |
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