Theorem nvop2 30538

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 30532	. . 3 ⊢ Rel NrmCVec
2		1st2nd 8045	. . 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 30537	. . 3 ⊢ 𝑁 = (2nd ‘𝑈)
7	4, 6	opeq12i 4876	. 2 ⊢ ⟨𝑊, 𝑁⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
8	3, 7	eqtr4di 2784	1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 = ⟨𝑊, 𝑁⟩)

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099  ⟨cop 4629  Rel wrel 5679  ‘cfv 6546  1^st c1st 7993  2^nd c2nd 7994  NrmCVeccnv 30514  norm_CVcnmcv 30520

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 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-iota 6498  df-fun 6548  df-fv 6554  df-oprab 7420  df-1st 7995  df-2nd 7996  df-nv 30522  df-nmcv 30530

This theorem is referenced by:  nvvop  30539  nvi  30544