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Theorem lsmelval2 20701
Description: Subspace sum membership in terms of a sum of 1-dim subspaces (atoms), which can be useful for treating subspaces as projective lattice elements. (Contributed by NM, 9-Aug-2014.)
Hypotheses
Ref Expression
lsmelval2.v 𝑉 = (Baseβ€˜π‘Š)
lsmelval2.s 𝑆 = (LSubSpβ€˜π‘Š)
lsmelval2.p βŠ• = (LSSumβ€˜π‘Š)
lsmelval2.n 𝑁 = (LSpanβ€˜π‘Š)
lsmelval2.w (πœ‘ β†’ π‘Š ∈ LMod)
lsmelval2.t (πœ‘ β†’ 𝑇 ∈ 𝑆)
lsmelval2.u (πœ‘ β†’ π‘ˆ ∈ 𝑆)
Assertion
Ref Expression
lsmelval2 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ (𝑋 ∈ 𝑉 ∧ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
Distinct variable groups:   𝑦,𝑧, βŠ•   𝑦,𝑇,𝑧   𝑦,π‘ˆ,𝑧   𝑦,𝑉,𝑧   𝑦,π‘Š,𝑧   𝑦,𝑋,𝑧   πœ‘,𝑦,𝑧
Allowed substitution hints:   𝑆(𝑦,𝑧)   𝑁(𝑦,𝑧)
Proof of Theorem lsmelval2
StepHypRef Expression
1 lsmelval2.w . . . . . 6 (πœ‘ β†’ π‘Š ∈ LMod)
2 lsmelval2.t . . . . . 6 (πœ‘ β†’ 𝑇 ∈ 𝑆)
3 lsmelval2.s . . . . . . 7 𝑆 = (LSubSpβ€˜π‘Š)
43lsssubg 20573 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝑆) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
51, 2, 4syl2anc 584 . . . . 5 (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
6 lsmelval2.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
73lsssubg 20573 . . . . . 6 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
81, 6, 7syl2anc 584 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
9 eqid 2732 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
10 lsmelval2.p . . . . . 6 βŠ• = (LSSumβ€˜π‘Š)
119, 10lsmelval 19519 . . . . 5 ((𝑇 ∈ (SubGrpβ€˜π‘Š) ∧ π‘ˆ ∈ (SubGrpβ€˜π‘Š)) β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦(+gβ€˜π‘Š)𝑧)))
125, 8, 11syl2anc 584 . . . 4 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦(+gβ€˜π‘Š)𝑧)))
131adantr 481 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘Š ∈ LMod)
142adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ 𝑆)
15 simprl 769 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑇)
16 lsmelval2.v . . . . . . . . . . . 12 𝑉 = (Baseβ€˜π‘Š)
1716, 3lssel 20553 . . . . . . . . . . 11 ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) β†’ 𝑦 ∈ 𝑉)
1814, 15, 17syl2anc 584 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑉)
19 lsmelval2.n . . . . . . . . . . 11 𝑁 = (LSpanβ€˜π‘Š)
2016, 3, 19lspsncl 20593 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
2113, 18, 20syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
223lsssubg 20573 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑦}) ∈ 𝑆) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
2313, 21, 22syl2anc 584 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
246adantr 481 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ 𝑆)
25 simprr 771 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ)
2616, 3lssel 20553 . . . . . . . . . . 11 ((π‘ˆ ∈ 𝑆 ∧ 𝑧 ∈ π‘ˆ) β†’ 𝑧 ∈ 𝑉)
2724, 25, 26syl2anc 584 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ 𝑉)
2816, 3, 19lspsncl 20593 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
2913, 27, 28syl2anc 584 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
303lsssubg 20573 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑧}) ∈ 𝑆) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3113, 29, 30syl2anc 584 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3216, 19lspsnid 20609 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3313, 18, 32syl2anc 584 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3416, 19lspsnid 20609 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
3513, 27, 34syl2anc 584 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
369, 10lsmelvali 19520 . . . . . . . 8 ((((π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š)) ∧ (𝑦 ∈ (π‘β€˜{𝑦}) ∧ 𝑧 ∈ (π‘β€˜{𝑧}))) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))
3723, 31, 33, 35, 36syl22anc 837 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))
38 eleq1a 2828 . . . . . . 7 ((𝑦(+gβ€˜π‘Š)𝑧) ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) β†’ (𝑋 = (𝑦(+gβ€˜π‘Š)𝑧) β†’ 𝑋 ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))))
3937, 38syl 17 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑋 = (𝑦(+gβ€˜π‘Š)𝑧) β†’ 𝑋 ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))))
403, 10lsmcl 20699 . . . . . . . 8 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑦}) ∈ 𝑆 ∧ (π‘β€˜{𝑧}) ∈ 𝑆) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) ∈ 𝑆)
4113, 21, 29, 40syl3anc 1371 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) ∈ 𝑆)
4216, 3, 19, 13, 41lspsnel6 20610 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑋 ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) ↔ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
4339, 42sylibd 238 . . . . 5 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑋 = (𝑦(+gβ€˜π‘Š)𝑧) β†’ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
4443reximdvva 3205 . . . 4 (πœ‘ β†’ (βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦(+gβ€˜π‘Š)𝑧) β†’ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
4512, 44sylbid 239 . . 3 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) β†’ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
465adantr 481 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
473, 19, 13, 14, 15lspsnel5a 20612 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) βŠ† 𝑇)
4810lsmless1 19530 . . . . . . . 8 ((𝑇 ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{𝑦}) βŠ† 𝑇) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• (π‘β€˜{𝑧})))
4946, 31, 47, 48syl3anc 1371 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• (π‘β€˜{𝑧})))
508adantr 481 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
513, 19, 13, 24, 25lspsnel5a 20612 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) βŠ† π‘ˆ)
5210lsmless2 19531 . . . . . . . 8 ((𝑇 ∈ (SubGrpβ€˜π‘Š) ∧ π‘ˆ ∈ (SubGrpβ€˜π‘Š) ∧ (π‘β€˜{𝑧}) βŠ† π‘ˆ) β†’ (𝑇 βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5346, 50, 51, 52syl3anc 1371 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑇 βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5449, 53sstrd 3992 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5554sseld 3981 . . . . 5 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑋 ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) β†’ 𝑋 ∈ (𝑇 βŠ• π‘ˆ)))
5642, 55sylbird 259 . . . 4 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))) β†’ 𝑋 ∈ (𝑇 βŠ• π‘ˆ)))
5756rexlimdvva 3211 . . 3 (πœ‘ β†’ (βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))) β†’ 𝑋 ∈ (𝑇 βŠ• π‘ˆ)))
5845, 57impbid 211 . 2 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
59 r19.42v 3190 . . . 4 (βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))) ↔ (𝑋 ∈ 𝑉 ∧ βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))))
6059rexbii 3094 . . 3 (βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))) ↔ βˆƒπ‘¦ ∈ 𝑇 (𝑋 ∈ 𝑉 ∧ βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))))
61 r19.42v 3190 . . 3 (βˆƒπ‘¦ ∈ 𝑇 (𝑋 ∈ 𝑉 ∧ βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))) ↔ (𝑋 ∈ 𝑉 ∧ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))))
6260, 61bitri 274 . 2 (βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))) ↔ (𝑋 ∈ 𝑉 ∧ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧}))))
6358, 62bitrdi 286 1 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ (𝑋 ∈ 𝑉 ∧ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ (π‘β€˜{𝑋}) βŠ† ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   βŠ† wss 3948  {csn 4628  β€˜cfv 6543  (class class class)co 7411  Basecbs 17146  +gcplusg 17199  SubGrpcsubg 19002  LSSumclsm 19504  LModclmod 20475  LSubSpclss 20547  LSpanclspn 20587
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-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  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-sets 17099  df-slot 17117  df-ndx 17129  df-base 17147  df-ress 17176  df-plusg 17212  df-0g 17389  df-mgm 18563  df-sgrp 18612  df-mnd 18628  df-submnd 18674  df-grp 18824  df-minusg 18825  df-sbg 18826  df-subg 19005  df-cntz 19183  df-lsm 19506  df-cmn 19652  df-abl 19653  df-mgp 19990  df-ur 20007  df-ring 20060  df-lmod 20477  df-lss 20548  df-lsp 20588
This theorem is referenced by:  dihjat1lem  40385
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