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Theorem nvvc 29906
Description: The vector space component of a normed complex vector space. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
nvvc.1 π‘Š = (1st β€˜π‘ˆ)
Assertion
Ref Expression
nvvc (π‘ˆ ∈ NrmCVec β†’ π‘Š ∈ CVecOLD)
Proof of Theorem nvvc
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nvvc.1 . . 3 π‘Š = (1st β€˜π‘ˆ)
2 eqid 2732 . . 3 ( +𝑣 β€˜π‘ˆ) = ( +𝑣 β€˜π‘ˆ)
3 eqid 2732 . . 3 ( ·𝑠OLD β€˜π‘ˆ) = ( ·𝑠OLD β€˜π‘ˆ)
41, 2, 3nvvop 29900 . 2 (π‘ˆ ∈ NrmCVec β†’ π‘Š = ⟨( +𝑣 β€˜π‘ˆ), ( ·𝑠OLD β€˜π‘ˆ)⟩)
5 eqid 2732 . . . 4 (BaseSetβ€˜π‘ˆ) = (BaseSetβ€˜π‘ˆ)
6 eqid 2732 . . . 4 (0vecβ€˜π‘ˆ) = (0vecβ€˜π‘ˆ)
7 eqid 2732 . . . 4 (normCVβ€˜π‘ˆ) = (normCVβ€˜π‘ˆ)
85, 2, 3, 6, 7nvi 29905 . . 3 (π‘ˆ ∈ NrmCVec β†’ (⟨( +𝑣 β€˜π‘ˆ), ( ·𝑠OLD β€˜π‘ˆ)⟩ ∈ CVecOLD ∧ (normCVβ€˜π‘ˆ):(BaseSetβ€˜π‘ˆ)βŸΆβ„ ∧ βˆ€π‘₯ ∈ (BaseSetβ€˜π‘ˆ)((((normCVβ€˜π‘ˆ)β€˜π‘₯) = 0 β†’ π‘₯ = (0vecβ€˜π‘ˆ)) ∧ βˆ€π‘¦ ∈ β„‚ ((normCVβ€˜π‘ˆ)β€˜(𝑦( ·𝑠OLD β€˜π‘ˆ)π‘₯)) = ((absβ€˜π‘¦) Β· ((normCVβ€˜π‘ˆ)β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ (BaseSetβ€˜π‘ˆ)((normCVβ€˜π‘ˆ)β€˜(π‘₯( +𝑣 β€˜π‘ˆ)𝑦)) ≀ (((normCVβ€˜π‘ˆ)β€˜π‘₯) + ((normCVβ€˜π‘ˆ)β€˜π‘¦)))))
98simp1d 1142 . 2 (π‘ˆ ∈ NrmCVec β†’ ⟨( +𝑣 β€˜π‘ˆ), ( ·𝑠OLD β€˜π‘ˆ)⟩ ∈ CVecOLD)
104, 9eqeltrd 2833 1 (π‘ˆ ∈ NrmCVec β†’ π‘Š ∈ CVecOLD)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βŸ¨cop 4634   class class class wbr 5148  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411  1st c1st 7975  β„‚cc 11110  β„cr 11111  0cc0 11112   + caddc 11115   Β· cmul 11117   ≀ cle 11251  abscabs 15183  CVecOLDcvc 29849  NrmCVeccnv 29875   +𝑣 cpv 29876  BaseSetcba 29877   ·𝑠OLD cns 29878  0veccn0v 29879  normCVcnmcv 29881
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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
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-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-1st 7977  df-2nd 7978  df-vc 29850  df-nv 29883  df-va 29886  df-ba 29887  df-sm 29888  df-0v 29889  df-nmcv 29891
This theorem is referenced by:  nvablo  29907  nvsf  29910  nvscl  29917  nvsid  29918  nvsass  29919  nvdi  29921  nvdir  29922  nv2  29923  nv0  29928  nvsz  29929  nvinv  29930  phop  30109  ip0i  30116  ipdirilem  30120  hlvc  30184
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