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Mirrors > Home > MPE Home > Th. List > hl0cl | Structured version Visualization version GIF version |
Description: The Hilbert space zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hl0cl.1 | β’ π = (BaseSetβπ) |
hl0cl.5 | β’ π = (0vecβπ) |
Ref | Expression |
---|---|
hl0cl | β’ (π β CHilOLD β π β π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 29294 | . 2 β’ (π β CHilOLD β π β NrmCVec) | |
2 | hl0cl.1 | . . 3 β’ π = (BaseSetβπ) | |
3 | hl0cl.5 | . . 3 β’ π = (0vecβπ) | |
4 | 2, 3 | nvzcl 29037 | . 2 β’ (π β NrmCVec β π β π) |
5 | 1, 4 | syl 17 | 1 β’ (π β CHilOLD β π β π) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1539 β wcel 2104 βcfv 6454 NrmCVeccnv 28987 BaseSetcba 28989 0veccn0v 28991 CHilOLDchlo 29288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-riota 7260 df-ov 7306 df-oprab 7307 df-1st 7859 df-2nd 7860 df-grpo 28896 df-gid 28897 df-ablo 28948 df-vc 28962 df-nv 28995 df-va 28998 df-ba 28999 df-sm 29000 df-0v 29001 df-nmcv 29003 df-cbn 29266 df-hlo 29289 |
This theorem is referenced by: axhv0cl-zf 29388 |
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