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| Mirrors > Home > MPE Home > Th. List > hladdid | Structured version Visualization version GIF version | ||
| Description: Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| hladdid.1 | β’ π = (BaseSetβπ) |
| hladdid.2 | β’ πΊ = ( +π£ βπ) |
| hladdid.5 | β’ π = (0vecβπ) |
| Ref | Expression |
|---|---|
| hladdid | β’ ((π β CHilOLD β§ π΄ β π) β (π΄πΊπ) = π΄) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | hlnv 30719 | . 2 β’ (π β CHilOLD β π β NrmCVec) | |
| 2 | hladdid.1 | . . 3 β’ π = (BaseSetβπ) | |
| 3 | hladdid.2 | . . 3 β’ πΊ = ( +π£ βπ) | |
| 4 | hladdid.5 | . . 3 β’ π = (0vecβπ) | |
| 5 | 2, 3, 4 | nv0rid 30463 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (π΄πΊπ) = π΄) |
| 6 | 1, 5 | sylan 578 | 1 β’ ((π β CHilOLD β§ π΄ β π) β (π΄πΊπ) = π΄) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 βcfv 6551 (class class class)co 7424 NrmCVeccnv 30412 +π£ cpv 30413 BaseSetcba 30414 0veccn0v 30416 CHilOLDchlo 30713 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
| This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-1st 7997 df-2nd 7998 df-grpo 30321 df-gid 30322 df-ablo 30373 df-vc 30387 df-nv 30420 df-va 30423 df-ba 30424 df-sm 30425 df-0v 30426 df-nmcv 30428 df-cbn 30691 df-hlo 30714 |
| This theorem is referenced by: axhvaddid-zf 30814 |
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