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Mirrors > Home > MPE Home > Th. List > isnvi | Structured version Visualization version GIF version |
Description: Properties that determine a normed complex vector space. (Contributed by NM, 15-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
isnvi.5 | β’ π = ran πΊ |
isnvi.6 | β’ π = (GIdβπΊ) |
isnvi.7 | β’ β¨πΊ, πβ© β CVecOLD |
isnvi.8 | β’ π:πβΆβ |
isnvi.9 | β’ ((π₯ β π β§ (πβπ₯) = 0) β π₯ = π) |
isnvi.10 | β’ ((π¦ β β β§ π₯ β π) β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯))) |
isnvi.11 | β’ ((π₯ β π β§ π¦ β π) β (πβ(π₯πΊπ¦)) β€ ((πβπ₯) + (πβπ¦))) |
isnvi.12 | β’ π = β¨β¨πΊ, πβ©, πβ© |
Ref | Expression |
---|---|
isnvi | β’ π β NrmCVec |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnvi.12 | . 2 β’ π = β¨β¨πΊ, πβ©, πβ© | |
2 | isnvi.7 | . . 3 β’ β¨πΊ, πβ© β CVecOLD | |
3 | isnvi.8 | . . 3 β’ π:πβΆβ | |
4 | isnvi.9 | . . . . . 6 β’ ((π₯ β π β§ (πβπ₯) = 0) β π₯ = π) | |
5 | 4 | ex 412 | . . . . 5 β’ (π₯ β π β ((πβπ₯) = 0 β π₯ = π)) |
6 | isnvi.10 | . . . . . . 7 β’ ((π¦ β β β§ π₯ β π) β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯))) | |
7 | 6 | ancoms 458 | . . . . . 6 β’ ((π₯ β π β§ π¦ β β) β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯))) |
8 | 7 | ralrimiva 3145 | . . . . 5 β’ (π₯ β π β βπ¦ β β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯))) |
9 | isnvi.11 | . . . . . 6 β’ ((π₯ β π β§ π¦ β π) β (πβ(π₯πΊπ¦)) β€ ((πβπ₯) + (πβπ¦))) | |
10 | 9 | ralrimiva 3145 | . . . . 5 β’ (π₯ β π β βπ¦ β π (πβ(π₯πΊπ¦)) β€ ((πβπ₯) + (πβπ¦))) |
11 | 5, 8, 10 | 3jca 1127 | . . . 4 β’ (π₯ β π β (((πβπ₯) = 0 β π₯ = π) β§ βπ¦ β β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯)) β§ βπ¦ β π (πβ(π₯πΊπ¦)) β€ ((πβπ₯) + (πβπ¦)))) |
12 | 11 | rgen 3062 | . . 3 β’ βπ₯ β π (((πβπ₯) = 0 β π₯ = π) β§ βπ¦ β β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯)) β§ βπ¦ β π (πβ(π₯πΊπ¦)) β€ ((πβπ₯) + (πβπ¦))) |
13 | isnvi.5 | . . . 4 β’ π = ran πΊ | |
14 | isnvi.6 | . . . 4 β’ π = (GIdβπΊ) | |
15 | 13, 14 | isnv 30298 | . . 3 β’ (β¨β¨πΊ, πβ©, πβ© β NrmCVec β (β¨πΊ, πβ© β CVecOLD β§ π:πβΆβ β§ βπ₯ β π (((πβπ₯) = 0 β π₯ = π) β§ βπ¦ β β (πβ(π¦ππ₯)) = ((absβπ¦) Β· (πβπ₯)) β§ βπ¦ β π (πβ(π₯πΊπ¦)) β€ ((πβπ₯) + (πβπ¦))))) |
16 | 2, 3, 12, 15 | mpbir3an 1340 | . 2 β’ β¨β¨πΊ, πβ©, πβ© β NrmCVec |
17 | 1, 16 | eqeltri 2828 | 1 β’ π β NrmCVec |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 βwral 3060 β¨cop 4634 class class class wbr 5148 ran crn 5677 βΆwf 6539 βcfv 6543 (class class class)co 7412 βcc 11114 βcr 11115 0cc0 11116 + caddc 11119 Β· cmul 11121 β€ 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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