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Theorem cnnvg 27763

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 2724	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2	1	vafval 27688	. 2 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
3		cnnvg.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6307	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5037	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 14197	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 10130	. . . . . 6 ⊢ ℂ ∈ V
8		fex 6605	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 710	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 7293	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2746	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6307	. 2 ⊢ (1^st ‘(1^st ‘𝑈)) = (1^st ‘⟨ + , · ⟩)
13		addex 11944	. . 3 ⊢ + ∈ V
14		mulex 11945	. . 3 ⊢ · ∈ V
15	13, 14	op1st 7293	. 2 ⊢ (1^st ‘⟨ + , · ⟩) = +
16	2, 12, 15	3eqtrri 2751	1 ⊢ + = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1596 ∈ wcel 2103 Vcvv 3304 ⟨cop 4291 ⟶wf 5997 ‘cfv 6001 1^st c1st 7283 ℂcc 10047 ℝcr 10048 + caddc 10052 · cmul 10054 abscabs 14094 +_𝑣 cpv 27670

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1835 ax-4 1850 ax-5 1952 ax-6 2018 ax-7 2054 ax-8 2105 ax-9 2112 ax-10 2132 ax-11 2147 ax-12 2160 ax-13 2355 ax-ext 2704 ax-rep 4879 ax-sep 4889 ax-nul 4897 ax-pow 4948 ax-pr 5011 ax-un 7066 ax-cnex 10105 ax-resscn 10106 ax-1cn 10107 ax-icn 10108 ax-addcl 10109 ax-addrcl 10110 ax-mulcl 10111 ax-mulrcl 10112 ax-mulcom 10113 ax-addass 10114 ax-mulass 10115 ax-distr 10116 ax-i2m1 10117 ax-1ne0 10118 ax-1rid 10119 ax-rnegex 10120 ax-rrecex 10121 ax-cnre 10122 ax-pre-lttri 10123 ax-pre-lttrn 10124 ax-pre-ltadd 10125 ax-pre-mulgt0 10126 ax-pre-sup 10127 ax-addf 10128 ax-mulf 10129

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1599 df-ex 1818 df-nf 1823 df-sb 2011 df-eu 2575 df-mo 2576 df-clab 2711 df-cleq 2717 df-clel 2720 df-nfc 2855 df-ne 2897 df-nel 3000 df-ral 3019 df-rex 3020 df-reu 3021 df-rmo 3022 df-rab 3023 df-v 3306 df-sbc 3542 df-csb 3640 df-dif 3683 df-un 3685 df-in 3687 df-ss 3694 df-pss 3696 df-nul 4024 df-if 4195 df-pw 4268 df-sn 4286 df-pr 4288 df-tp 4290 df-op 4292 df-uni 4545 df-iun 4630 df-br 4761 df-opab 4821 df-mpt 4838 df-tr 4861 df-id 5128 df-eprel 5133 df-po 5139 df-so 5140 df-fr 5177 df-we 5179 df-xp 5224 df-rel 5225 df-cnv 5226 df-co 5227 df-dm 5228 df-rn 5229 df-res 5230 df-ima 5231 df-pred 5793 df-ord 5839 df-on 5840 df-lim 5841 df-suc 5842 df-iota 5964 df-fun 6003 df-fn 6004 df-f 6005 df-f1 6006 df-fo 6007 df-f1o 6008 df-fv 6009 df-riota 6726 df-ov 6768 df-oprab 6769 df-mpt2 6770 df-om 7183 df-1st 7285 df-2nd 7286 df-wrecs 7527 df-recs 7588 df-rdg 7626 df-er 7862 df-en 8073 df-dom 8074 df-sdom 8075 df-sup 8464 df-pnf 10189 df-mnf 10190 df-xr 10191 df-ltxr 10192 df-le 10193 df-sub 10381 df-neg 10382 df-div 10798 df-nn 11134 df-2 11192 df-3 11193 df-n0 11406 df-z 11491 df-uz 11801 df-rp 11947 df-seq 12917 df-exp 12976 df-cj 13959 df-re 13960 df-im 13961 df-sqrt 14095 df-abs 14096 df-va 27680

This theorem is referenced by: cnnvba 27764 cnnvm 27767 ipblnfi 27941

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