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Theorem obselocv 21283
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 2733 . . . . . . 7 (0gβ€˜π‘Š) = (0gβ€˜π‘Š)
21obsne0 21280 . . . . . 6 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 β‰  (0gβ€˜π‘Š))
323adant2 1132 . . . . 5 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 β‰  (0gβ€˜π‘Š))
4 elin 3965 . . . . . . . 8 (𝐴 ∈ (𝐢 ∩ ( βŠ₯ β€˜πΆ)) ↔ (𝐴 ∈ 𝐢 ∧ 𝐴 ∈ ( βŠ₯ β€˜πΆ)))
5 obsrcl 21278 . . . . . . . . . . . . . 14 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ π‘Š ∈ PreHil)
653ad2ant1 1134 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ π‘Š ∈ PreHil)
7 phllmod 21183 . . . . . . . . . . . . 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 1138 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐢 βŠ† 𝐡)
10 eqid 2733 . . . . . . . . . . . . . . 15 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
1110obsss 21279 . . . . . . . . . . . . . 14 (𝐡 ∈ (OBasisβ€˜π‘Š) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
12113ad2ant1 1134 . . . . . . . . . . . . 13 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐡 βŠ† (Baseβ€˜π‘Š))
139, 12sstrd 3993 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐢 βŠ† (Baseβ€˜π‘Š))
14 eqid 2733 . . . . . . . . . . . . 13 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
1510, 14lspssid 20596 . . . . . . . . . . . 12 ((π‘Š ∈ LMod ∧ 𝐢 βŠ† (Baseβ€˜π‘Š)) β†’ 𝐢 βŠ† ((LSpanβ€˜π‘Š)β€˜πΆ))
168, 13, 15syl2anc 585 . . . . . . . . . . 11 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐢 βŠ† ((LSpanβ€˜π‘Š)β€˜πΆ))
1716ssrind 4236 . . . . . . . . . 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 21229 . . . . . . . . . . . . 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 585 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ ( βŠ₯ β€˜((LSpanβ€˜π‘Š)β€˜πΆ)) = ( βŠ₯ β€˜πΆ))
2120ineq2d 4213 . . . . . . . . . . 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 2733 . . . . . . . . . . . . . 14 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
2310, 22, 14lspcl 20587 . . . . . . . . . . . . 13 ((π‘Š ∈ LMod ∧ 𝐢 βŠ† (Baseβ€˜π‘Š)) β†’ ((LSpanβ€˜π‘Š)β€˜πΆ) ∈ (LSubSpβ€˜π‘Š))
248, 13, 23syl2anc 585 . . . . . . . . . . . 12 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ ((LSpanβ€˜π‘Š)β€˜πΆ) ∈ (LSubSpβ€˜π‘Š))
2518, 22, 1ocvin 21227 . . . . . . . . . . . 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 585 . . . . . . . . . . 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 2775 . . . . . . . . . 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 4023 . . . . . . . . 9 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ (𝐢 ∩ ( βŠ₯ β€˜πΆ)) βŠ† {(0gβ€˜π‘Š)})
2928sseld 3982 . . . . . . . 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 4646 . . . . . . 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 2954 . . . . 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 401 . . . 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 486 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐢)
39 eleq1 2822 . . . . . . 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 1192 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ 𝐡 ∈ (OBasisβ€˜π‘Š))
43 simpl3 1194 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ 𝐴 ∈ 𝐡)
449sselda 3983 . . . . . . 7 (((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) ∧ π‘₯ ∈ 𝐢) β†’ π‘₯ ∈ 𝐡)
45 eqid 2733 . . . . . . . 8 (Β·π‘–β€˜π‘Š) = (Β·π‘–β€˜π‘Š)
46 eqid 2733 . . . . . . . 8 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
47 eqid 2733 . . . . . . . 8 (1rβ€˜(Scalarβ€˜π‘Š)) = (1rβ€˜(Scalarβ€˜π‘Š))
48 eqid 2733 . . . . . . . 8 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
4910, 45, 46, 47, 48obsip 21276 . . . . . . 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 1372 . . . . . 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 4538 . . . . . . 7 (Β¬ 𝐴 = π‘₯ β†’ if(𝐴 = π‘₯, (1rβ€˜(Scalarβ€˜π‘Š)), (0gβ€˜(Scalarβ€˜π‘Š))) = (0gβ€˜(Scalarβ€˜π‘Š)))
5251eqeq2d 2744 . . . . . 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 3155 . . 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 1139 . . . . 5 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ 𝐡)
5712, 56sseldd 3984 . . . 4 ((𝐡 ∈ (OBasisβ€˜π‘Š) ∧ 𝐢 βŠ† 𝐡 ∧ 𝐴 ∈ 𝐡) β†’ 𝐴 ∈ (Baseβ€˜π‘Š))
5810, 45, 46, 48, 18elocv 21221 . . . . . 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 1090 . . . . . 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 275 . . . . 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 537 . . . 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 585 . . 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 259 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062   ∩ cin 3948   βŠ† wss 3949  ifcif 4529  {csn 4629  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200  Β·π‘–cip 17202  0gc0g 17385  1rcur 20004  LModclmod 20471  LSubSpclss 20542  LSpanclspn 20582  PreHilcphl 21177  ocvcocv 21213  OBasiscobs 21257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-mulr 17211  df-sca 17213  df-vsca 17214  df-ip 17215  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-mhm 18671  df-grp 18822  df-minusg 18823  df-sbg 18824  df-ghm 19090  df-mgp 19988  df-ur 20005  df-ring 20058  df-oppr 20150  df-dvdsr 20171  df-unit 20172  df-rnghom 20251  df-drng 20359  df-staf 20453  df-srng 20454  df-lmod 20473  df-lss 20543  df-lsp 20583  df-lmhm 20633  df-lvec 20714  df-sra 20785  df-rgmod 20786  df-phl 21179  df-ocv 21216  df-obs 21260
This theorem is referenced by:  obs2ss  21284  obslbs  21285
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