Theorem nvop2 30417

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 30411	. . 3 ⊢ Rel NrmCVec
2		1st2nd 8043	. . 3 ⊢ ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
3	1, 2	mpan 689	. 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 30416	. . 3 ⊢ 𝑁 = (2nd ‘𝑈)
7	4, 6	opeq12i 4879	. 2 ⊢ ⟨𝑊, 𝑁⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
8	3, 7	eqtr4di 2786	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ⟨cop 4635  Rel wrel 5683  ‘cfv 6548  1^st c1st 7991  2^nd c2nd 7992  NrmCVeccnv 30393  norm_CVcnmcv 30399

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6500  df-fun 6550  df-fv 6556  df-oprab 7424  df-1st 7993  df-2nd 7994  df-nv 30401  df-nmcv 30409

This theorem is referenced by:  nvvop  30418  nvi  30423