Theorem nvop2 30594

Description: A normed complex vector space is an ordered pair of a vector space and a norm operation. (Contributed by NM, 28-Nov-2006.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvop2.1	⊢ 𝑊 = (1st ‘𝑈)
nvop2.6	⊢ 𝑁 = (normCV‘𝑈)

Assertion

Ref	Expression
nvop2	⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Proof of Theorem nvop2

Step	Hyp	Ref	Expression
1		nvrel 30588	. . 3 ⊢ Rel NrmCVec
2		1st2nd 8043	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
3	1, 2	mpan 690	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
4		nvop2.1	. . 3 ⊢ 𝑊 = (1st ‘𝑈)
5		nvop2.6	. . . 4 ⊢ 𝑁 = (normCV‘𝑈)
6	5	nmcvfval 30593	. . 3 ⊢ 𝑁 = (2nd ‘𝑈)
7	4, 6	opeq12i 4859	. 2 ⊢ ⟨𝑊, 𝑁⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
8	3, 7	eqtr4di 2789	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4612  Rel wrel 5664  ‘cfv 6536  1^st c1st 7991  2^nd c2nd 7992  NrmCVeccnv 30570  norm_CVcnmcv 30576

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 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fv 6544  df-oprab 7414  df-1st 7993  df-2nd 7994  df-nv 30578  df-nmcv 30586

This theorem is referenced by:  nvvop  30595  nvi  30600