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Theorem isnvi 30299
Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
isnvi.5 𝑋 = ran 𝐺
isnvi.6 𝑍 = (GIdβ€˜πΊ)
isnvi.7 ⟨𝐺, π‘†βŸ© ∈ CVecOLD
isnvi.8 𝑁:π‘‹βŸΆβ„
isnvi.9 ((π‘₯ ∈ 𝑋 ∧ (π‘β€˜π‘₯) = 0) β†’ π‘₯ = 𝑍)
isnvi.10 ((𝑦 ∈ β„‚ ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)))
isnvi.11 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
isnvi.12 π‘ˆ = ⟨⟨𝐺, π‘†βŸ©, π‘βŸ©
Assertion
Ref Expression
isnvi π‘ˆ ∈ NrmCVec
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯,𝑁,𝑦   π‘₯,𝑆,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(π‘₯,𝑦)   𝑍(π‘₯,𝑦)

Proof of Theorem isnvi
StepHypRef Expression
1 isnvi.12 . 2 π‘ˆ = ⟨⟨𝐺, π‘†βŸ©, π‘βŸ©
2 isnvi.7 . . 3 ⟨𝐺, π‘†βŸ© ∈ CVecOLD
3 isnvi.8 . . 3 𝑁:π‘‹βŸΆβ„
4 isnvi.9 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ (π‘β€˜π‘₯) = 0) β†’ π‘₯ = 𝑍)
54ex 412 . . . . 5 (π‘₯ ∈ 𝑋 β†’ ((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍))
6 isnvi.10 . . . . . . 7 ((𝑦 ∈ β„‚ ∧ π‘₯ ∈ 𝑋) β†’ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)))
76ancoms 458 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ β„‚) β†’ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)))
87ralrimiva 3145 . . . . 5 (π‘₯ ∈ 𝑋 β†’ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)))
9 isnvi.11 . . . . . 6 ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
109ralrimiva 3145 . . . . 5 (π‘₯ ∈ 𝑋 β†’ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
115, 8, 103jca 1127 . . . 4 (π‘₯ ∈ 𝑋 β†’ (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))
1211rgen 3062 . . 3 βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))
13 isnvi.5 . . . 4 𝑋 = ran 𝐺
14 isnvi.6 . . . 4 𝑍 = (GIdβ€˜πΊ)
1513, 14isnv 30298 . . 3 (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ∧ βˆ€π‘¦ ∈ β„‚ (π‘β€˜(𝑦𝑆π‘₯)) = ((absβ€˜π‘¦) Β· (π‘β€˜π‘₯)) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯𝐺𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
162, 3, 12, 15mpbir3an 1340 . 2 ⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec
171, 16eqeltri 2828 1 π‘ˆ ∈ NrmCVec
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βŸ¨cop 4634   class class class wbr 5148  ran crn 5677  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11114  β„cr 11115  0cc0 11116   + caddc 11119   Β· cmul 11121   ≀ cle 11256  abscabs 15188  GIdcgi 30176  CVecOLDcvc 30244  NrmCVeccnv 30270
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7415  df-oprab 7416  df-vc 30245  df-nv 30278
This theorem is referenced by:  cnnv  30363  hhnv  30851  hhssnv  30950
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