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 30489

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 2728	. . 3 ⊢ ( ·_𝑠OLD ‘𝑈) = ( ·_𝑠OLD ‘𝑈)
2	1	smfval 30414	. 2 ⊢ ( ·_𝑠OLD ‘𝑈) = (2^nd ‘(1^st ‘𝑈))
3		cnnvs.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6900	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5466	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 15316	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 11219	. . . . . 6 ⊢ ℂ ∈ V
8		fex 7238	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 691	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 8001	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2756	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6900	. 2 ⊢ (2^nd ‘(1^st ‘𝑈)) = (2^nd ‘⟨ + , · ⟩)
13		addex 13003	. . 3 ⊢ + ∈ V
14		mulex 13005	. . 3 ⊢ · ∈ V
15	13, 14	op2nd 8002	. 2 ⊢ (2^nd ‘⟨ + , · ⟩) = ·
16	2, 12, 15	3eqtrri 2761	1 ⊢ · = ( ·_𝑠OLD ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1534 ∈ wcel 2099 Vcvv 3471 ⟨cop 4635 ⟶wf 6544 ‘cfv 6548 1^st c1st 7991 2^nd c2nd 7992 ℂcc 11136 ℝcr 11137 + caddc 11141 · cmul 11143 abscabs 15213 ·_𝑠OLD cns 30396

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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-sm 30406

This theorem is referenced by: cnnvm 30491 ipblnfi 30664

W3C validator