Theorem nvop2 29848

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 29842	. . 3 ⊢ Rel NrmCVec
2		1st2nd 8021	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
3	1, 2	mpan 688	. 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 29847	. . 3 ⊢ 𝑁 = (2nd ‘𝑈)
7	4, 6	opeq12i 4877	. 2 ⊢ ⟨𝑊, 𝑁⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
8	3, 7	eqtr4di 2790	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ⟨cop 4633  Rel wrel 5680  ‘cfv 6540  1^st c1st 7969  2^nd c2nd 7970  NrmCVeccnv 29824  norm_CVcnmcv 29830

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fv 6548  df-oprab 7409  df-1st 7971  df-2nd 7972  df-nv 29832  df-nmcv 29840

This theorem is referenced by:  nvvop  29849  nvi  29854