cnnvs - Metamath Proof Explorer

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Theorem cnnvs 28943

Description: The scalar product operation of the normed complex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (New usage is discouraged.)

**Hypothesis**
Ref	Expression
cnnvs.6	⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩

**Assertion**
Ref	Expression
cnnvs	⊢ · = ( ·_𝑠OLD ‘𝑈)

**Proof of Theorem cnnvs**
Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
2	1	smfval 28868	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈))
3		cnnvs.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6759	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5373	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 14977	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 10883	. . . . . 6 ⊢ ℂ ∈ V
8		fex 7084	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 688	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 7812	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2766	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6759	. 2 ⊢ (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘⟨ + , · ⟩)
13		addex 12657	. . 3 ⊢ + ∈ V
14		mulex 12658	. . 3 ⊢ · ∈ V
15	13, 14	op2nd 7813	. 2 ⊢ (2^nd ‘⟨ + , · ⟩) = ·
16	2, 12, 15	3eqtrri 2771	1 ⊢ · = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2108 Vcvv 3422 ⟨cop 4564 ⟶wf 6414 ‘cfv 6418 1^st c1st 7802 2^nd c2nd 7803 ℂcc 10800 ℝcr 10801 + caddc 10805 · cmul 10807 abscabs 14873 ·_𝑠OLD cns 28850

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-rep 5205 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-addf 10881 ax-mulf 10882

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-1st 7804 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-sm 28860

This theorem is referenced by: cnnvm 28945 ipblnfi 29118

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