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Theorem obselocv 21289
Description: A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypothesis
Ref Expression
obselocv.o βŠ₯ = (ocvβ€˜π‘Š)
Assertion
Ref Expression
obselocv ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 ∈ ( βŠ₯ β€˜πΆ) ↔ Β¬ 𝐴 ∈ 𝐢))
Proof of Theorem obselocv
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 eqid 2732 . . . . . . 7 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
21obsne0 21286 . . . . . 6 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 β‰  (0gβ€˜π‘Š))
323adant2 1131 . . . . 5 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 β‰  (0gβ€˜π‘Š))
4 elin 3964 . . . . . . . 8 (𝐴 ∈ (𝐢 ∩ ( βŠ₯ β€˜πΆ)) ↔ (𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ)))
5 obsrcl 21284 . . . . . . . . . . . . . 14 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ PreHil)
653ad2ant1 1133 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ π‘Š ∈ PreHil)
7 phllmod 21189 . . . . . . . . . . . . 13 (π‘Š ∈ PreHil β†’ π‘Š ∈ LMod)
86, 7syl 17 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ π‘Š ∈ LMod)
9 simp2 1137 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐢 βŠ† 𝐡)
10 eqid 2732 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
1110obsss 21285 . . . . . . . . . . . . . 14 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
12113ad2ant1 1133 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
139, 12sstrd 3992 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐢 βŠ† (Baseβ€˜π‘Š))
14 eqid 2732 . . . . . . . . . . . . 13 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
1510, 14lspssid 20601 . . . . . . . . . . . 12 ((π‘Š ∈ LMod ∧ 𝐢 βŠ† (Baseβ€˜π‘Š)) β†’ 𝐢 βŠ† ((LSpanβ€˜π‘Š)β€˜πΆ))
168, 13, 15syl2anc 584 . . . . . . . . . . 11 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐢 βŠ† ((LSpanβ€˜π‘Š)β€˜πΆ))
1716ssrind 4235 . . . . . . . . . 10 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐢 ∩ ( βŠ₯ β€˜πΆ)) βŠ† (((LSpanβ€˜π‘Š)β€˜πΆ) ∩ ( βŠ₯ β€˜πΆ)))
18 obselocv.o . . . . . . . . . . . . . 14 βŠ₯ = (ocvβ€˜π‘Š)
1910, 18, 14ocvlsp 21235 . . . . . . . . . . . . 13 ((π‘Š ∈ PreHil ∧ 𝐢 βŠ† (Baseβ€˜π‘Š)) β†’ ( βŠ₯ β€˜((LSpanβ€˜π‘Š)β€˜πΆ)) = ( βŠ₯ β€˜πΆ))
206, 13, 19syl2anc 584 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ ( βŠ₯ β€˜((LSpanβ€˜π‘Š)β€˜πΆ)) = ( βŠ₯ β€˜πΆ))
2120ineq2d 4212 . . . . . . . . . . 11 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (((LSpanβ€˜π‘Š)β€˜πΆ) ∩ ( βŠ₯ β€˜((LSpanβ€˜π‘Š)β€˜πΆ))) = (((LSpanβ€˜π‘Š)β€˜πΆ) ∩ ( βŠ₯ β€˜πΆ)))
22 eqid 2732 . . . . . . . . . . . . . 14 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
2310, 22, 14lspcl 20592 . . . . . . . . . . . . 13 ((π‘Š ∈ LMod ∧ 𝐢 βŠ† (Baseβ€˜π‘Š)) β†’ ((LSpanβ€˜π‘Š)β€˜πΆ) ∈ (LSubSpβ€˜π‘Š))
248, 13, 23syl2anc 584 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ ((LSpanβ€˜π‘Š)β€˜πΆ) ∈ (LSubSpβ€˜π‘Š))
2518, 22, 1ocvin 21233 . . . . . . . . . . . 12 ((π‘Š ∈ PreHil ∧ ((LSpanβ€˜π‘Š)β€˜πΆ) ∈ (LSubSpβ€˜π‘Š)) β†’ (((LSpanβ€˜π‘Š)β€˜πΆ) ∩ ( βŠ₯ β€˜((LSpanβ€˜π‘Š)β€˜πΆ))) = {(0gβ€˜π‘Š)})
266, 24, 25syl2anc 584 . . . . . . . . . . 11 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (((LSpanβ€˜π‘Š)β€˜πΆ) ∩ ( βŠ₯ β€˜((LSpanβ€˜π‘Š)β€˜πΆ))) = {(0gβ€˜π‘Š)})
2721, 26eqtr3d 2774 . . . . . . . . . 10 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (((LSpanβ€˜π‘Š)β€˜πΆ) ∩ ( βŠ₯ β€˜πΆ)) = {(0gβ€˜π‘Š)})
2817, 27sseqtrd 4022 . . . . . . . . 9 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐢 ∩ ( βŠ₯ β€˜πΆ)) βŠ† {(0gβ€˜π‘Š)})
2928sseld 3981 . . . . . . . 8 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 ∈ (𝐢 ∩ ( βŠ₯ β€˜πΆ)) β†’ 𝐴 ∈ {(0gβ€˜π‘Š)}))
304, 29biimtrrid 242 . . . . . . 7 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ ((𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ)) β†’ 𝐴 ∈ {(0gβ€˜π‘Š)}))
31 elsni 4645 . . . . . . 7 (𝐴 ∈ {(0gβ€˜π‘Š)} β†’ 𝐴 = (0gβ€˜π‘Š))
3230, 31syl6 35 . . . . . 6 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ ((𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ)) β†’ 𝐴 = (0gβ€˜π‘Š)))
3332necon3ad 2953 . . . . 5 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 β‰  (0gβ€˜π‘Š) β†’ Β¬ (𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ))))
343, 33mpd 15 . . . 4 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ Β¬ (𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ)))
35 imnan 400 . . . 4 ((𝐴 ∈ 𝐢 β†’ Β¬ 𝐴 ∈ ( βŠ₯ β€˜πΆ)) ↔ Β¬ (𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ)))
3634, 35sylibr 233 . . 3 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 ∈ 𝐢 β†’ Β¬ 𝐴 ∈ ( βŠ₯ β€˜πΆ)))
3736con2d 134 . 2 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 ∈ ( βŠ₯ β€˜πΆ) β†’ Β¬ 𝐴 ∈ 𝐢))
38 simpr 485 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐢)
39 eleq1 2821 . . . . . . 7 (𝐴 = π‘₯ β†’ (𝐴 ∈ 𝐢 ↔ π‘₯ ∈ 𝐢))
4038, 39syl5ibrcom 246 . . . . . 6 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ (𝐴 = π‘₯ β†’ 𝐴 ∈ 𝐢))
4140con3d 152 . . . . 5 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ (Β¬ 𝐴 ∈ 𝐢 β†’ Β¬ 𝐴 = π‘₯))
42 simpl1 1191 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ 𝐡 ∈ (OBasisβ€˜π‘Š))
43 simpl3 1193 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ 𝐴 ∈ 𝐡)
449sselda 3982 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐡)
45 eqid 2732 . . . . . . . 8 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
46 eqid 2732 . . . . . . . 8 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
47 eqid 2732 . . . . . . . 8 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
48 eqid 2732 . . . . . . . 8 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
4910, 45, 46, 47, 48obsip 21282 . . . . . . 7 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡 ∧ π‘₯ ∈ 𝐡) β†’ (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = if(𝐴 = π‘₯, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))))
5042, 43, 44, 49syl3anc 1371 . . . . . 6 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = if(𝐴 = π‘₯, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))))
51 iffalse 4537 . . . . . . 7 (Β¬ 𝐴 = π‘₯ β†’ if(𝐴 = π‘₯, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
5251eqeq2d 2743 . . . . . 6 (Β¬ 𝐴 = π‘₯ β†’ ((𝐴(Β·π‘–β€˜π‘Š)π‘₯) = if(𝐴 = π‘₯, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))) ↔ (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
5350, 52syl5ibcom 244 . . . . 5 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ (Β¬ 𝐴 = π‘₯ β†’ (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
5441, 53syld 47 . . . 4 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ (Β¬ 𝐴 ∈ 𝐢 β†’ (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
5554ralrimdva 3154 . . 3 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (Β¬ 𝐴 ∈ 𝐢 β†’ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
56 simp3 1138 . . . . 5 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ 𝐡)
5712, 56sseldd 3983 . . . 4 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ (Baseβ€˜π‘Š))
5810, 45, 46, 48, 18elocv 21227 . . . . . 6 (𝐴 ∈ ( βŠ₯ β€˜πΆ) ↔ (𝐢 βŠ† (Baseβ€˜π‘Š) ∧ 𝐴 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
59 df-3an 1089 . . . . . 6 ((𝐢 βŠ† (Baseβ€˜π‘Š) ∧ 𝐴 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))) ↔ ((𝐢 βŠ† (Baseβ€˜π‘Š) ∧ 𝐴 ∈ (Baseβ€˜π‘Š)) ∧ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
6058, 59bitri 274 . . . . 5 (𝐴 ∈ ( βŠ₯ β€˜πΆ) ↔ ((𝐢 βŠ† (Baseβ€˜π‘Š) ∧ 𝐴 ∈ (Baseβ€˜π‘Š)) ∧ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
6160baib 536 . . . 4 ((𝐢 βŠ† (Baseβ€˜π‘Š) ∧ 𝐴 ∈ (Baseβ€˜π‘Š)) β†’ (𝐴 ∈ ( βŠ₯ β€˜πΆ) ↔ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
6213, 57, 61syl2anc 584 . . 3 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 ∈ ( βŠ₯ β€˜πΆ) ↔ βˆ€π‘₯ ∈ 𝐢 (𝐴(Β·π‘–β€˜π‘Š)π‘₯) = (0gβ€˜(Scalarβ€˜π‘Š))))
6355, 62sylibrd 258 . 2 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (Β¬ 𝐴 ∈ 𝐢 β†’ 𝐴 ∈ ( βŠ₯ β€˜πΆ)))
6437, 63impbid 211 1 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐴 ∈ ( βŠ₯ β€˜πΆ) ↔ Β¬ 𝐴 ∈ 𝐢))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061   ∩ cin 3947   βŠ† wss 3948  ifcif 4528  {csn 4628  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  Scalarcsca 17202  Β·π‘–cip 17204  0gc0g 17387  1rcur 20006  LModclmod 20475  LSubSpclss 20547  LSpanclspn 20587  PreHilcphl 21183  ocvcocv 21219  OBasiscobs 21263
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 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  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 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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  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 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-2 12277  df-3 12278  df-4 12279  df-5 12280  df-6 12281  df-7 12282  df-8 12283  df-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-plusg 17212  df-mulr 17213  df-sca 17215  df-vsca 17216  df-ip 17217  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-mhm 18673  df-grp 18824  df-minusg 18825  df-sbg 18826  df-ghm 19092  df-mgp 19990  df-ur 20007  df-ring 20060  df-oppr 20154  df-dvdsr 20175  df-unit 20176  df-rnghom 20255  df-drng 20363  df-staf 20457  df-srng 20458  df-lmod 20477  df-lss 20548  df-lsp 20588  df-lmhm 20638  df-lvec 20719  df-sra 20791  df-rgmod 20792  df-phl 21185  df-ocv 21222  df-obs 21266
This theorem is referenced by:  obs2ss  21290  obslbs  21291
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