cnnvg - Metamath Proof Explorer

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Theorem cnnvg 28222

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

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

**Assertion**
Ref	Expression
cnnvg	⊢ + = ( +_𝑣 ‘𝑈)

**Proof of Theorem cnnvg**
Step	Hyp	Ref	Expression
1		eqid 2772	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2	1	vafval 28147	. 2 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
3		cnnvg.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6496	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5206	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 14548	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 10408	. . . . . 6 ⊢ ℂ ∈ V
8		fex 6809	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 679	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 7502	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2796	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6496	. 2 ⊢ (1^st ‘(1^st ‘𝑈)) = (1^st ‘⟨ + , · ⟩)
13		addex 12195	. . 3 ⊢ + ∈ V
14		mulex 12196	. . 3 ⊢ · ∈ V
15	13, 14	op1st 7502	. 2 ⊢ (1^st ‘⟨ + , · ⟩) = +
16	2, 12, 15	3eqtrri 2801	1 ⊢ + = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1507 ∈ wcel 2048 Vcvv 3409 ⟨cop 4441 ⟶wf 6178 ‘cfv 6182 1^st c1st 7492 ℂcc 10325 ℝcr 10326 + caddc 10330 · cmul 10332 abscabs 14444 +_𝑣 cpv 28129

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-sup 8693 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-z 11787 df-uz 12052 df-rp 12198 df-seq 13178 df-exp 13238 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-va 28139

This theorem is referenced by: cnnvba 28223 cnnvm 28226 ipblnfi 28400

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