cnnvg - Metamath Proof Explorer

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Theorem cnnvg 30737

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 2735	. . 3 ⊢ ( +_𝑣 ‘𝑈) = ( +_𝑣 ‘𝑈)
2	1	vafval 30662	. 2 ⊢ ( +_𝑣 ‘𝑈) = (1^st ‘(1^st ‘𝑈))
3		cnnvg.6	. . . . 5 ⊢ 𝑈 = ⟨⟨ + , · ⟩, abs⟩
4	3	fveq2i 6832	. . . 4 ⊢ (1^st ‘𝑈) = (1^st ‘⟨⟨ + , · ⟩, abs⟩)
5		opex 5405	. . . . 5 ⊢ ⟨ + , · ⟩ ∈ V
6		absf 15289	. . . . . 6 ⊢ abs:ℂ⟶ℝ
7		cnex 11108	. . . . . 6 ⊢ ℂ ∈ V
8		fex 7170	. . . . . 6 ⊢ ((abs:ℂ⟶ℝ ∧ ℂ ∈ V) → abs ∈ V)
9	6, 7, 8	mp2an 693	. . . . 5 ⊢ abs ∈ V
10	5, 9	op1st 7939	. . . 4 ⊢ (1^st ‘⟨⟨ + , · ⟩, abs⟩) = ⟨ + , · ⟩
11	4, 10	eqtri 2758	. . 3 ⊢ (1^st ‘𝑈) = ⟨ + , · ⟩
12	11	fveq2i 6832	. 2 ⊢ (1^st ‘(1^st ‘𝑈)) = (1^st ‘⟨ + , · ⟩)
13		addex 12928	. . 3 ⊢ + ∈ V
14		mulex 12930	. . 3 ⊢ · ∈ V
15	13, 14	op1st 7939	. 2 ⊢ (1^st ‘⟨ + , · ⟩) = +
16	2, 12, 15	3eqtrri 2763	1 ⊢ + = ( +_𝑣 ‘𝑈)

Colors of variables: wff setvar class

Syntax hints: = wceq 1542 ∈ wcel 2114 Vcvv 3427 ⟨cop 4563 ⟶wf 6483 ‘cfv 6487 1^st c1st 7929 ℂcc 11025 ℝcr 11026 + caddc 11030 · cmul 11032 abscabs 15185 +_𝑣 cpv 30644

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3060 df-rmo 3340 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6254 df-ord 6315 df-on 6316 df-lim 6317 df-suc 6318 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8632 df-en 8883 df-dom 8884 df-sdom 8885 df-sup 9344 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12164 df-2 12233 df-3 12234 df-n0 12427 df-z 12514 df-uz 12778 df-rp 12932 df-seq 13953 df-exp 14013 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-va 30654

This theorem is referenced by: cnnvba 30738 cnnvm 30741 ipblnfi 30914

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