cnnvg - Metamath Proof Explorer

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Theorem cnnvg 30657

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 2729	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2	1	vafval 30582	. 2 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
3		cnnvg.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6843	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5419	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 15280	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 11125	. . . . . 6 ⊢ ℂ ∈ V
8		fex 7182	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 692	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 7955	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2752	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6843	. 2 ⊢ (1^st ‘(1^st ‘𝑈)) = (1^st ‘⟨ + , · ⟩)
13		addex 12924	. . 3 ⊢ + ∈ V
14		mulex 12926	. . 3 ⊢ · ∈ V
15	13, 14	op1st 7955	. 2 ⊢ (1^st ‘⟨ + , · ⟩) = +
16	2, 12, 15	3eqtrri 2757	1 ⊢ + = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1540 ∈ wcel 2109 Vcvv 3444 ⟨cop 4591 ⟶wf 6495 ‘cfv 6499 1^st c1st 7945 ℂcc 11042 ℝcr 11043 + caddc 11047 · cmul 11049 abscabs 15176 +_𝑣 cpv 30564

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-1st 7947 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-sup 9369 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-seq 13943 df-exp 14003 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-va 30574

This theorem is referenced by: cnnvba 30658 cnnvm 30661 ipblnfi 30834

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