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Theorem nvop 27855
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop.2 𝐺 = ( +𝑣𝑈)
nvop.4 𝑆 = ( ·𝑠OLD𝑈)
nvop.6 𝑁 = (normCV𝑈)
Assertion
Ref Expression
nvop (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)

Proof of Theorem nvop
StepHypRef Expression
1 nvrel 27781 . . 3 Rel NrmCVec
2 1st2nd 7442 . . 3 ((Rel NrmCVec ∧ 𝑈 ∈ NrmCVec) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
31, 2mpan 673 . 2 (𝑈 ∈ NrmCVec → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
4 nvop.6 . . . . 5 𝑁 = (normCV𝑈)
54nmcvfval 27786 . . . 4 𝑁 = (2nd𝑈)
65opeq2i 4599 . . 3 ⟨(1st𝑈), 𝑁⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
7 eqid 2806 . . . . 5 (1st𝑈) = (1st𝑈)
8 nvop.2 . . . . 5 𝐺 = ( +𝑣𝑈)
9 nvop.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
107, 8, 9nvvop 27788 . . . 4 (𝑈 ∈ NrmCVec → (1st𝑈) = ⟨𝐺, 𝑆⟩)
1110opeq1d 4601 . . 3 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), 𝑁⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
126, 11syl5eqr 2854 . 2 (𝑈 ∈ NrmCVec → ⟨(1st𝑈), (2nd𝑈)⟩ = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
133, 12eqtrd 2840 1 (𝑈 ∈ NrmCVec → 𝑈 = ⟨⟨𝐺, 𝑆⟩, 𝑁⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2156  cop 4376  Rel wrel 5316  cfv 6097  1st c1st 7392  2nd c2nd 7393  NrmCVeccnv 27763   +𝑣 cpv 27764   ·𝑠OLD cns 27766  normCVcnmcv 27769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-fo 6103  df-fv 6105  df-oprab 6874  df-1st 7394  df-2nd 7395  df-vc 27738  df-nv 27771  df-va 27774  df-sm 27776  df-nmcv 27779
This theorem is referenced by:  sspval  27902  isph  28001  hilhhi  28345
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