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Theorem obslbs 21276
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 21269 . . . . . 6 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ PreHil)
2 eqid 2732 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
32obsss 21270 . . . . . 6 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
4 eqid 2732 . . . . . . 7 (ocvβ€˜π‘Š) = (ocvβ€˜π‘Š)
5 obslbs.n . . . . . . 7 𝑁 = (LSpanβ€˜π‘Š)
62, 4, 5ocvlsp 21220 . . . . . 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 584 . . . . 5 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅)) = ((ocvβ€˜π‘Š)β€˜π΅))
87fveq2d 6892 . . . 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 21273 . . . 4 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜π΅)) = (Baseβ€˜π‘Š))
108, 9eqtrd 2772 . . 3 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜((ocvβ€˜π‘Š)β€˜(π‘β€˜π΅))) = (Baseβ€˜π‘Š))
1110eqeq2d 2743 . 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 21227 . . 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 21173 . . . 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 482 . . . . . 6 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) β†’ βˆƒπ‘¦(𝑦 ∈ 𝐡 ∧ Β¬ 𝑦 ∈ π‘₯))
19 simpll 765 . . . . . . . . . . 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 725 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘₯ βŠ† 𝐡)
22 simpr 485 . . . . . . . . . . 11 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 ∈ 𝐡)
234obselocv 21274 . . . . . . . . . . 11 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ βŠ† 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) ↔ Β¬ 𝑦 ∈ π‘₯))
2419, 21, 22, 23syl3anc 1371 . . . . . . . . . 10 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) ↔ Β¬ 𝑦 ∈ π‘₯))
25 eqid 2732 . . . . . . . . . . . . . 14 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
2625obsne0 21271 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝑦 ∈ 𝐡) β†’ 𝑦 β‰  (0gβ€˜π‘Š))
2719, 22, 26syl2anc 584 . . . . . . . . . . . 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 3039 . . . . . . . . . . . 12 ((𝑦 ∈ ((ocvβ€˜π‘Š)β€˜π‘₯) ∧ Β¬ 𝑦 ∈ {(0gβ€˜π‘Š)}) β†’ ((ocvβ€˜π‘Š)β€˜π‘₯) β‰  {(0gβ€˜π‘Š)})
3130expcom 414 . . . . . . . . . . 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 21174 . . . . . . . . . . . . . . 15 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
361, 35syl 17 . . . . . . . . . . . . . 14 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ LMod)
3736ad2antrr 724 . . . . . . . . . . . . 13 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘Š ∈ LMod)
383ad2antrr 724 . . . . . . . . . . . . . 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 20586 . . . . . . . . . . . . 13 ((π‘Š ∈ LMod ∧ π‘₯ βŠ† (Baseβ€˜π‘Š)) β†’ (π‘β€˜π‘₯) βŠ† (Baseβ€˜π‘Š))
4137, 39, 40syl2anc 584 . . . . . . . . . . . 12 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ (π‘β€˜π‘₯) βŠ† (Baseβ€˜π‘Š))
42 fveq2 6888 . . . . . . . . . . . . 13 ((π‘β€˜π‘₯) = (Baseβ€˜π‘Š) β†’ ((ocvβ€˜π‘Š)β€˜(π‘β€˜π‘₯)) = ((ocvβ€˜π‘Š)β€˜(Baseβ€˜π‘Š)))
431ad2antrr 724 . . . . . . . . . . . . . . 15 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) ∧ 𝑦 ∈ 𝐡) β†’ π‘Š ∈ PreHil)
442, 4, 5ocvlsp 21220 . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . 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 21223 . . . . . . . . . . . . . . 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 2748 . . . . . . . . . . . . 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 2957 . . . . . . . . 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 454 . . . . . . 7 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ π‘₯ ⊊ 𝐡) β†’ ((𝑦 ∈ 𝐡 ∧ Β¬ 𝑦 ∈ π‘₯) β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5554exlimdv 1936 . . . . . 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 413 . . . 4 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ (π‘₯ ⊊ 𝐡 β†’ (π‘β€˜π‘₯) ⊊ (Baseβ€˜π‘Š)))
5857alrimiv 1930 . . 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 20760 . . . . 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 1097 . . . . 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 540 . . 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 835 . 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 396   ∧ w3a 1087  βˆ€wal 1539   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106   β‰  wne 2940   βŠ† wss 3947   ⊊ wpss 3948  {csn 4627  β€˜cfv 6540  Basecbs 17140  0gc0g 17381  LModclmod 20463  LSpanclspn 20574  LBasisclbs 20677  LVecclvec 20705  PreHilcphl 21168  ocvcocv 21204  ClSubSpccss 21205  OBasiscobs 21248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-tpos 8207  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-mulr 17207  df-sca 17209  df-vsca 17210  df-ip 17211  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-mhm 18667  df-grp 18818  df-minusg 18819  df-sbg 18820  df-ghm 19084  df-mgp 19982  df-ur 19999  df-ring 20051  df-oppr 20142  df-dvdsr 20163  df-unit 20164  df-invr 20194  df-rnghom 20243  df-drng 20309  df-staf 20445  df-srng 20446  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lmhm 20625  df-lbs 20678  df-lvec 20706  df-sra 20777  df-rgmod 20778  df-phl 21170  df-ocv 21207  df-css 21208  df-obs 21251
This theorem is referenced by: (None)
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