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Theorem obslbs 21504
Description: An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
Hypotheses
Ref Expression
obslbs.j 𝐽 = (LBasisβ€˜π‘Š)
obslbs.n 𝑁 = (LSpanβ€˜π‘Š)
obslbs.c 𝐢 = (ClSubSpβ€˜π‘Š)
Assertion
Ref Expression
obslbs (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (𝐡 ∈ 𝐽 ↔ (π‘β€˜π΅) ∈ 𝐢))
Proof of Theorem obslbs
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 obsrcl 21497 . . . . . 6 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ PreHil)
2 eqid 2730 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
32obsss 21498 . . . . . 6 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
4 eqid 2730 . . . . . . 7 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
5 obslbs.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘Š)
62, 4, 5ocvlsp 21448 . . . . . 6 ((π‘Š ∈ PreHil ∧ 𝐡 βŠ† (Baseβ€˜π‘Š)) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅)) = ((ocvβ€˜π‘Š)β€˜π΅))
71, 3, 6syl2anc 582 . . . . 5 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅)) = ((ocvβ€˜π‘Š)β€˜π΅))
87fveq2d 6894 . . . 4 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅))) = ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π΅)))
94, 2obs2ocv 21501 . . . 4 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π΅)) = (Baseβ€˜π‘Š))
108, 9eqtrd 2770 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅))) = (Baseβ€˜π‘Š))
1110eqeq2d 2741 . 2 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((π‘β€˜π΅) = ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅))) ↔ (π‘β€˜π΅) = (Baseβ€˜π‘Š)))
12 obslbs.c . . . 4 𝐢 = (ClSubSpβ€˜π‘Š)
134, 12iscss 21455 . . 3 (π‘Š ∈ PreHil β†’ ((π‘β€˜π΅) ∈ 𝐢 ↔ (π‘β€˜π΅) = ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅)))))
141, 13syl 17 . 2 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((π‘β€˜π΅) ∈ 𝐢 ↔ (π‘β€˜π΅) = ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅)))))
15 phllvec 21401 . . . 4 (π‘Š ∈ PreHil β†’ π‘Š ∈ LVec)
161, 15syl 17 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ LVec)
17 pssnel 4469 . . . . . . 7 (π‘₯ ⊊ 𝐡 β†’ βˆƒπ‘¦(𝑦 ∈ 𝐡 ∧ Β¬ 𝑦 ∈ π‘₯))
1817adantl 480 . . . . . 6 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) β†’ βˆƒπ‘¦(𝑦 ∈ 𝐡 ∧ Β¬ 𝑦 ∈ π‘₯))
19 simpll 763 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ 𝐡 ∈ (OBasisβ€˜π‘Š))
20 pssss 4094 . . . . . . . . . . . 12 (π‘₯ ⊊ 𝐡 β†’ π‘₯ βŠ† 𝐡)
2120ad2antlr 723 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ βŠ† 𝐡)
22 simpr 483 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 ∈ 𝐡)
234obselocv 21502 . . . . . . . . . . 11 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) ↔ Β¬ 𝑦 ∈ π‘₯))
2419, 21, 22, 23syl3anc 1369 . . . . . . . . . 10 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) ↔ Β¬ 𝑦 ∈ π‘₯))
25 eqid 2730 . . . . . . . . . . . . . 14 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
2625obsne0 21499 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 β‰  (0gβ€˜π‘Š))
2719, 22, 26syl2anc 582 . . . . . . . . . . . 12 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 β‰  (0gβ€˜π‘Š))
28 nelsn 4667 . . . . . . . . . . . 12 (𝑦 β‰  (0gβ€˜π‘Š) β†’ Β¬ 𝑦 ∈ {(0gβ€˜π‘Š)})
2927, 28syl 17 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ Β¬ 𝑦 ∈ {(0gβ€˜π‘Š)})
30 nelne1 3037 . . . . . . . . . . . 12 ((𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) ∧ Β¬ 𝑦 ∈ {(0gβ€˜π‘Š)}) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) β‰  {(0gβ€˜π‘Š)})
3130expcom 412 . . . . . . . . . . 11 (Β¬ 𝑦 ∈ {(0gβ€˜π‘Š)} β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) β‰  {(0gβ€˜π‘Š)}))
3229, 31syl 17 . . . . . . . . . 10 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) β‰  {(0gβ€˜π‘Š)}))
3324, 32sylbird 259 . . . . . . . . 9 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (Β¬ 𝑦 ∈ π‘₯ β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) β‰  {(0gβ€˜π‘Š)}))
34 npss 4109 . . . . . . . . . . 11 (Β¬ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š) ↔ ((π‘β€˜π‘₯) βŠ† (Baseβ€˜π‘Š) β†’ (π‘β€˜π‘₯) = (Baseβ€˜π‘Š)))
35 phllmod 21402 . . . . . . . . . . . . . . 15 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
361, 35syl 17 . . . . . . . . . . . . . 14 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ LMod)
3736ad2antrr 722 . . . . . . . . . . . . 13 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘Š ∈ LMod)
383ad2antrr 722 . . . . . . . . . . . . . 14 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
3921, 38sstrd 3991 . . . . . . . . . . . . 13 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ βŠ† (Baseβ€˜π‘Š))
402, 5lspssv 20738 . . . . . . . . . . . . 13 ((π‘Š ∈ LMod ∧ π‘₯ βŠ† (Baseβ€˜π‘Š)) β†’ (π‘β€˜π‘₯) βŠ† (Baseβ€˜π‘Š))
4137, 39, 40syl2anc 582 . . . . . . . . . . . 12 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (π‘β€˜π‘₯) βŠ† (Baseβ€˜π‘Š))
42 fveq2 6890 . . . . . . . . . . . . 13 ((π‘β€˜π‘₯) = (Baseβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π‘₯)) = ((ocvβ€˜π‘Š)β€˜(Baseβ€˜π‘Š)))
431ad2antrr 722 . . . . . . . . . . . . . . 15 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘Š ∈ PreHil)
442, 4, 5ocvlsp 21448 . . . . . . . . . . . . . . 15 ((π‘Š ∈ PreHil ∧ π‘₯ βŠ† (Baseβ€˜π‘Š)) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π‘₯)) = ((ocvβ€˜π‘Š)β€˜π‘₯))
4543, 39, 44syl2anc 582 . . . . . . . . . . . . . 14 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π‘₯)) = ((ocvβ€˜π‘Š)β€˜π‘₯))
462, 4, 25ocv1 21451 . . . . . . . . . . . . . . 15 (π‘Š ∈ PreHil β†’ ((ocvβ€˜π‘Š)β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)})
4743, 46syl 17 . . . . . . . . . . . . . 14 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ ((ocvβ€˜π‘Š)β€˜(Baseβ€˜π‘Š)) = {(0gβ€˜π‘Š)})
4845, 47eqeq12d 2746 . . . . . . . . . . . . 13 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (((ocvβ€˜π‘Š)β€˜(π‘β€˜π‘₯)) = ((ocvβ€˜π‘Š)β€˜(Baseβ€˜π‘Š)) ↔ ((ocvβ€˜π‘Š)β€˜π‘₯) = {(0gβ€˜π‘Š)}))
4942, 48imbitrid 243 . . . . . . . . . . . 12 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ ((π‘β€˜π‘₯) = (Baseβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) = {(0gβ€˜π‘Š)}))
5041, 49embantd 59 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (((π‘β€˜π‘₯) βŠ† (Baseβ€˜π‘Š) β†’ (π‘β€˜π‘₯) = (Baseβ€˜π‘Š)) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) = {(0gβ€˜π‘Š)}))
5134, 50biimtrid 241 . . . . . . . . . 10 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (Β¬ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) = {(0gβ€˜π‘Š)}))
5251necon1ad 2955 . . . . . . . . 9 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (((ocvβ€˜π‘Š)β€˜π‘₯) β‰  {(0gβ€˜π‘Š)} β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5333, 52syld 47 . . . . . . . 8 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (Β¬ 𝑦 ∈ π‘₯ β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5453expimpd 452 . . . . . . 7 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) β†’ ((𝑦 ∈ 𝐡 ∧ Β¬ 𝑦 ∈ π‘₯) β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5554exlimdv 1934 . . . . . 6 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) β†’ (βˆƒπ‘¦(𝑦 ∈ 𝐡 ∧ Β¬ 𝑦 ∈ π‘₯) β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5618, 55mpd 15 . . . . 5 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š))
5756ex 411 . . . 4 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5857alrimiv 1928 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ βˆ€π‘₯(π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
59 obslbs.j . . . . . 6 𝐽 = (LBasisβ€˜π‘Š)
602, 59, 5islbs3 20913 . . . . 5 (π‘Š ∈ LVec β†’ (𝐡 ∈ 𝐽 ↔ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ (π‘β€˜π΅) = (Baseβ€˜π‘Š) ∧ βˆ€π‘₯(π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))))
61 3anan32 1095 . . . . 5 ((𝐡 βŠ† (Baseβ€˜π‘Š) ∧ (π‘β€˜π΅) = (Baseβ€˜π‘Š) ∧ βˆ€π‘₯(π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š))) ↔ ((𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘₯(π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š))) ∧ (π‘β€˜π΅) = (Baseβ€˜π‘Š)))
6260, 61bitrdi 286 . . . 4 (π‘Š ∈ LVec β†’ (𝐡 ∈ 𝐽 ↔ ((𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘₯(π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š))) ∧ (π‘β€˜π΅) = (Baseβ€˜π‘Š))))
6362baibd 538 . . 3 ((π‘Š ∈ LVec ∧ (𝐡 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘₯(π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))) β†’ (𝐡 ∈ 𝐽 ↔ (π‘β€˜π΅) = (Baseβ€˜π‘Š)))
6416, 3, 58, 63syl12anc 833 . 2 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (𝐡 ∈ 𝐽 ↔ (π‘β€˜π΅) = (Baseβ€˜π‘Š)))
6511, 14, 643bitr4rd 311 1 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (𝐡 ∈ 𝐽 ↔ (π‘β€˜π΅) ∈ 𝐢))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085  βˆ€wal 1537   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938   βŠ† wss 3947   ⊊ wpss 3948  {csn 4627  β€˜cfv 6542  Basecbs 17148  0gc0g 17389  LModclmod 20614  LSpanclspn 20726  LBasisclbs 20829  LVecclvec 20857  PreHilcphl 21396  ocvcocv 21432  ClSubSpccss 21433  OBasiscobs 21476
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 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-tpos 8213  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-mhm 18705  df-grp 18858  df-minusg 18859  df-sbg 18860  df-ghm 19128  df-cmn 19691  df-abl 19692  df-mgp 20029  df-rng 20047  df-ur 20076  df-ring 20129  df-oppr 20225  df-dvdsr 20248  df-unit 20249  df-invr 20279  df-rhm 20363  df-drng 20502  df-staf 20596  df-srng 20597  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lmhm 20777  df-lbs 20830  df-lvec 20858  df-sra 20930  df-rgmod 20931  df-phl 21398  df-ocv 21435  df-css 21436  df-obs 21479
This theorem is referenced by: (None)
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