cnnvs - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cnnvs		Structured version Visualization version GIF version

Theorem cnnvs 30610

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 2726	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
2	1	smfval 30535	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈))
3		cnnvs.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6896	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5462	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 15337	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 11230	. . . . . 6 ⊢ ℂ ∈ V
8		fex 7235	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 690	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 8003	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2754	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6896	. 2 ⊢ (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘⟨ + , · ⟩)
13		addex 13019	. . 3 ⊢ + ∈ V
14		mulex 13021	. . 3 ⊢ · ∈ V
15	13, 14	op2nd 8004	. 2 ⊢ (2^nd ‘⟨ + , · ⟩) = ·
16	2, 12, 15	3eqtrri 2759	1 ⊢ · = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⟨cop 4629 ⟶wf 6542 ‘cfv 6546 1^st c1st 7993 2^nd c2nd 7994 ℂcc 11147 ℝcr 11148 + caddc 11152 · cmul 11154 abscabs 15234 ·_𝑠OLD cns 30517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228 ax-mulf 11229

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-sm 30527

This theorem is referenced by: cnnvm 30612 ipblnfi 30785

W3C validator