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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 30649 | . 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 30393 | . 2 β’ ((π β NrmCVec β§ π΄ β π) β (π΄πΊπ) = π΄) |
| 6 | 1, 5 | sylan 579 | 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 395 = wceq 1533 β wcel 2098 βcfv 6536 (class class class)co 7404 NrmCVeccnv 30342 +π£ cpv 30343 BaseSetcba 30344 0veccn0v 30346 CHilOLDchlo 30643 |
| 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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-1st 7971 df-2nd 7972 df-grpo 30251 df-gid 30252 df-ablo 30303 df-vc 30317 df-nv 30350 df-va 30353 df-ba 30354 df-sm 30355 df-0v 30356 df-nmcv 30358 df-cbn 30621 df-hlo 30644 |
| This theorem is referenced by: axhvaddid-zf 30744 |
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