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Theorem lsmelval2 20974
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 20845 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝑆) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
51, 2, 4syl2anc 582 . . . . 5 (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
6 lsmelval2.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
73lsssubg 20845 . . . . . 6 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
81, 6, 7syl2anc 582 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
9 eqid 2725 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
10 lsmelval2.p . . . . . 6 βŠ• = (LSSumβ€˜π‘Š)
119, 10lsmelval 19608 . . . . 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 582 . . . 4 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦(+gβ€˜π‘Š)𝑧)))
131adantr 479 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘Š ∈ LMod)
142adantr 479 . . . . . . . . . . 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 20825 . . . . . . . . . . 11 ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) β†’ 𝑦 ∈ 𝑉)
1814, 15, 17syl2anc 582 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑉)
19 lsmelval2.n . . . . . . . . . . 11 𝑁 = (LSpanβ€˜π‘Š)
2016, 3, 19lspsncl 20865 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
2113, 18, 20syl2anc 582 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
223lsssubg 20845 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑦}) ∈ 𝑆) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
2313, 21, 22syl2anc 582 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
246adantr 479 . . . . . . . . . . 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 20825 . . . . . . . . . . 11 ((π‘ˆ ∈ 𝑆 ∧ 𝑧 ∈ π‘ˆ) β†’ 𝑧 ∈ 𝑉)
2724, 25, 26syl2anc 582 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ 𝑉)
2816, 3, 19lspsncl 20865 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
2913, 27, 28syl2anc 582 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
303lsssubg 20845 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑧}) ∈ 𝑆) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3113, 29, 30syl2anc 582 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3216, 19lspsnid 20881 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3313, 18, 32syl2anc 582 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3416, 19lspsnid 20881 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
3513, 27, 34syl2anc 582 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
369, 10lsmelvali 19609 . . . . . . . 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 2820 . . . . . . 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 20972 . . . . . . . 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 1368 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) ∈ 𝑆)
4216, 3, 19, 13, 41lspsnel6 20882 . . . . . 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 3196 . . . 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 479 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
473, 19, 13, 14, 15lspsnel5a 20884 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) βŠ† 𝑇)
4810lsmless1 19619 . . . . . . . 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 1368 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• (π‘β€˜{𝑧})))
508adantr 479 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
513, 19, 13, 24, 25lspsnel5a 20884 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) βŠ† π‘ˆ)
5210lsmless2 19620 . . . . . . . 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 1368 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑇 βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5449, 53sstrd 3983 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5554sseld 3971 . . . . 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 3202 . . 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 3181 . . . 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 3084 . . 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 3181 . . 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 394   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060   βŠ† wss 3939  {csn 4624  β€˜cfv 6543  (class class class)co 7416  Basecbs 17179  +gcplusg 17232  SubGrpcsubg 19079  LSSumclsm 19593  LModclmod 20747  LSubSpclss 20819  LSpanclspn 20859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  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 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-sets 17132  df-slot 17150  df-ndx 17162  df-base 17180  df-ress 17209  df-plusg 17245  df-0g 17422  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-submnd 18740  df-grp 18897  df-minusg 18898  df-sbg 18899  df-subg 19082  df-cntz 19272  df-lsm 19595  df-cmn 19741  df-abl 19742  df-mgp 20079  df-ur 20126  df-ring 20179  df-lmod 20749  df-lss 20820  df-lsp 20860
This theorem is referenced by:  dihjat1lem  40957
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