Theorem nvop2 30683

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 30677	. . 3 ⊢ Rel NrmCVec
2		1st2nd 7983	. . 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 30682	. . 3 ⊢ 𝑁 = (2nd ‘𝑈)
7	4, 6	opeq12i 4834	. 2 ⊢ ⟨𝑊, 𝑁⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
8	3, 7	eqtr4di 2789	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ⟨cop 4586  Rel wrel 5629  ‘cfv 6492  1^st c1st 7931  2^nd c2nd 7932  NrmCVeccnv 30659  norm_CVcnmcv 30665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-oprab 7362  df-1st 7933  df-2nd 7934  df-nv 30667  df-nmcv 30675

This theorem is referenced by:  nvvop  30684  nvi  30689