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Theorem norm-iii-i 29402

Description: Theorem 3.3(iii) of [Beran] p. 97. (Contributed by NM, 29-Jul-1999.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
norm-iii.1	⊢ 𝐴 ∈ ℂ
norm-iii.2	⊢ 𝐵 ∈ ℋ

**Assertion**
Ref	Expression
norm-iii-i	⊢ (norm_ℎ‘(𝐴 ·_ℎ 𝐵)) = ((abs‘𝐴) · (norm_ℎ‘𝐵))

**Proof of Theorem norm-iii-i**
Step	Hyp	Ref	Expression
1		norm-iii.1	. . . . 5 ⊢ 𝐴 ∈ ℂ
2		norm-iii.2	. . . . 5 ⊢ 𝐵 ∈ ℋ
3	1, 1, 2, 2	his35i 29352	. . . 4 ⊢ ((𝐴 ·_ℎ 𝐵) ·_ih (𝐴 ·_ℎ 𝐵)) = ((𝐴 · (∗‘𝐴)) · (𝐵 ·_ih 𝐵))
4	3	fveq2i 6759	. . 3 ⊢ (√‘((𝐴 ·_ℎ 𝐵) ·_ih (𝐴 ·_ℎ 𝐵))) = (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·_ih 𝐵)))
5	1	cjmulrcli 14816	. . . 4 ⊢ (𝐴 · (∗‘𝐴)) ∈ ℝ
6		hiidrcl 29358	. . . . 5 ⊢ (𝐵 ∈ ℋ → (𝐵 ·_ih 𝐵) ∈ ℝ)
7	2, 6	ax-mp 5	. . . 4 ⊢ (𝐵 ·_ih 𝐵) ∈ ℝ
8	1	cjmulge0i 14818	. . . 4 ⊢ 0 ≤ (𝐴 · (∗‘𝐴))
9		hiidge0 29361	. . . . 5 ⊢ (𝐵 ∈ ℋ → 0 ≤ (𝐵 ·_ih 𝐵))
10	2, 9	ax-mp 5	. . . 4 ⊢ 0 ≤ (𝐵 ·_ih 𝐵)
11	5, 7, 8, 10	sqrtmulii 15026	. . 3 ⊢ (√‘((𝐴 · (∗‘𝐴)) · (𝐵 ·_ih 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·_ih 𝐵)))
12	4, 11	eqtri 2766	. 2 ⊢ (√‘((𝐴 ·_ℎ 𝐵) ·_ih (𝐴 ·_ℎ 𝐵))) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·_ih 𝐵)))
13	1, 2	hvmulcli 29277	. . 3 ⊢ (𝐴 ·_ℎ 𝐵) ∈ ℋ
14		normval 29387	. . 3 ⊢ ((𝐴 ·_ℎ 𝐵) ∈ ℋ → (norm_ℎ‘(𝐴 ·_ℎ 𝐵)) = (√‘((𝐴 ·_ℎ 𝐵) ·_ih (𝐴 ·_ℎ 𝐵))))
15	13, 14	ax-mp 5	. 2 ⊢ (norm_ℎ‘(𝐴 ·_ℎ 𝐵)) = (√‘((𝐴 ·_ℎ 𝐵) ·_ih (𝐴 ·_ℎ 𝐵)))
16		absval 14877	. . . 4 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴))))
17	1, 16	ax-mp 5	. . 3 ⊢ (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))
18		normval 29387	. . . 4 ⊢ (𝐵 ∈ ℋ → (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵)))
19	2, 18	ax-mp 5	. . 3 ⊢ (norm_ℎ‘𝐵) = (√‘(𝐵 ·_ih 𝐵))
20	17, 19	oveq12i 7267	. 2 ⊢ ((abs‘𝐴) · (norm_ℎ‘𝐵)) = ((√‘(𝐴 · (∗‘𝐴))) · (√‘(𝐵 ·_ih 𝐵)))
21	12, 15, 20	3eqtr4i 2776	1 ⊢ (norm_ℎ‘(𝐴 ·_ℎ 𝐵)) = ((abs‘𝐴) · (norm_ℎ‘𝐵))

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 · cmul 10807 ≤ cle 10941 ∗ccj 14735 √csqrt 14872 abscabs 14873 ℋchba 29182 ·_ℎ csm 29184 ·_ih csp 29185 norm_ℎcno 29186

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880 ax-hv0cl 29266 ax-hfvmul 29268 ax-hvmul0 29273 ax-hfi 29342 ax-his1 29345 ax-his3 29347 ax-his4 29348

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-hnorm 29231

This theorem is referenced by: norm-iii 29403 normsubi 29404 normpar2i 29419

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