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Theorem nvi 29867
Description: The properties of a normed complex vector space, which is a vector space accompanied by a norm. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 21-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvi.1 𝑋 = (BaseSetβ€˜π‘ˆ)
nvi.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
nvi.4 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
nvi.5 𝑍 = (0vecβ€˜π‘ˆ)
nvi.6 𝑁 = (normCVβ€˜π‘ˆ)
Assertion
Ref Expression
nvi (π‘ˆ ∈ NrmCVec β†’ (⟨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‘¦)))))
Distinct variable groups:   π‘₯,𝑦,𝐺   π‘₯,𝑁,𝑦   π‘₯,π‘ˆ   π‘₯,𝑆,𝑦   π‘₯,𝑋,𝑦
Allowed substitution hints:   π‘ˆ(𝑦)   𝑍(π‘₯,𝑦)

Proof of Theorem nvi
StepHypRef Expression
1 eqid 2733 . . . . . 6 (1st β€˜π‘ˆ) = (1st β€˜π‘ˆ)
2 nvi.6 . . . . . 6 𝑁 = (normCVβ€˜π‘ˆ)
31, 2nvop2 29861 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ = ⟨(1st β€˜π‘ˆ), π‘βŸ©)
4 nvi.2 . . . . . . 7 𝐺 = ( +𝑣 β€˜π‘ˆ)
5 nvi.4 . . . . . . 7 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
61, 4, 5nvvop 29862 . . . . . 6 (π‘ˆ ∈ NrmCVec β†’ (1st β€˜π‘ˆ) = ⟨𝐺, π‘†βŸ©)
76opeq1d 4880 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ ⟨(1st β€˜π‘ˆ), π‘βŸ© = ⟨⟨𝐺, π‘†βŸ©, π‘βŸ©)
83, 7eqtrd 2773 . . . 4 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ = ⟨⟨𝐺, π‘†βŸ©, π‘βŸ©)
9 id 22 . . . 4 (π‘ˆ ∈ NrmCVec β†’ π‘ˆ ∈ NrmCVec)
108, 9eqeltrrd 2835 . . 3 (π‘ˆ ∈ NrmCVec β†’ ⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec)
11 nvi.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
1211, 4bafval 29857 . . . 4 𝑋 = ran 𝐺
13 eqid 2733 . . . 4 (GIdβ€˜πΊ) = (GIdβ€˜πΊ)
1412, 13isnv 29865 . . 3 (⟨⟨𝐺, π‘†βŸ©, π‘βŸ© ∈ NrmCVec ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜πΊ)) ∧ 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‘¦)))))
1510, 14sylib 217 . 2 (π‘ˆ ∈ NrmCVec β†’ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜πΊ)) ∧ 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‘¦)))))
16 nvi.5 . . . . . . . 8 𝑍 = (0vecβ€˜π‘ˆ)
174, 160vfval 29859 . . . . . . 7 (π‘ˆ ∈ NrmCVec β†’ 𝑍 = (GIdβ€˜πΊ))
1817eqeq2d 2744 . . . . . 6 (π‘ˆ ∈ NrmCVec β†’ (π‘₯ = 𝑍 ↔ π‘₯ = (GIdβ€˜πΊ)))
1918imbi2d 341 . . . . 5 (π‘ˆ ∈ NrmCVec β†’ (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = 𝑍) ↔ ((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜πΊ))))
20193anbi1d 1441 . . . 4 (π‘ˆ ∈ NrmCVec 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‘¦))) ↔ (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜πΊ)) ∧ 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‘¦)))))
2120ralbidv 3178 . . 3 (π‘ˆ ∈ NrmCVec β†’ (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‘¦))) ↔ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜πΊ)) ∧ 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‘¦)))))
22213anbi3d 1443 . 2 (π‘ˆ ∈ NrmCVec β†’ ((⟨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‘¦)))) ↔ (⟨𝐺, π‘†βŸ© ∈ CVecOLD ∧ 𝑁:π‘‹βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 β†’ π‘₯ = (GIdβ€˜πΊ)) ∧ 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‘¦))))))
2315, 22mpbird 257 1 (π‘ˆ ∈ NrmCVec β†’ (⟨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‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βŸ¨cop 4635   class class class wbr 5149  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  1st c1st 7973  β„‚cc 11108  β„cr 11109  0cc0 11110   + caddc 11113   Β· cmul 11115   ≀ cle 11249  abscabs 15181  GIdcgi 29743  CVecOLDcvc 29811  NrmCVeccnv 29837   +𝑣 cpv 29838  BaseSetcba 29839   ·𝑠OLD cns 29840  0veccn0v 29841  normCVcnmcv 29843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-1st 7975  df-2nd 7976  df-vc 29812  df-nv 29845  df-va 29848  df-ba 29849  df-sm 29850  df-0v 29851  df-nmcv 29853
This theorem is referenced by:  nvvc  29868  nvf  29913  nvs  29916  nvz  29922  nvtri  29923
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