cnnvs - Metamath Proof Explorer

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Theorem cnnvs 30700

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 2736	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
2	1	smfval 30625	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈))
3		cnnvs.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6908	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5468	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 15377	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 11237	. . . . . 6 ⊢ ℂ ∈ V
8		fex 7247	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 692	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 8023	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2764	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6908	. 2 ⊢ (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘⟨ + , · ⟩)
13		addex 13032	. . 3 ⊢ + ∈ V
14		mulex 13034	. . 3 ⊢ · ∈ V
15	13, 14	op2nd 8024	. 2 ⊢ (2^nd ‘⟨ + , · ⟩) = ·
16	2, 12, 15	3eqtrri 2769	1 ⊢ · = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⟨cop 4631 ⟶wf 6556 ‘cfv 6560 1^st c1st 8013 2^nd c2nd 8014 ℂcc 11154 ℝcr 11155 + caddc 11159 · cmul 11161 abscabs 15274 ·_𝑠OLD cns 30607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-seq 14044 df-exp 14104 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-sm 30617

This theorem is referenced by: cnnvm 30702 ipblnfi 30875

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