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Theorem lsmelval2 20933
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 20804 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝑆) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
51, 2, 4syl2anc 583 . . . . 5 (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
6 lsmelval2.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
73lsssubg 20804 . . . . . 6 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
81, 6, 7syl2anc 583 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
9 eqid 2726 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
10 lsmelval2.p . . . . . 6 βŠ• = (LSSumβ€˜π‘Š)
119, 10lsmelval 19569 . . . . 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 583 . . . 4 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦(+gβ€˜π‘Š)𝑧)))
131adantr 480 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘Š ∈ LMod)
142adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ 𝑆)
15 simprl 768 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑇)
16 lsmelval2.v . . . . . . . . . . . 12 𝑉 = (Baseβ€˜π‘Š)
1716, 3lssel 20784 . . . . . . . . . . 11 ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) β†’ 𝑦 ∈ 𝑉)
1814, 15, 17syl2anc 583 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑉)
19 lsmelval2.n . . . . . . . . . . 11 𝑁 = (LSpanβ€˜π‘Š)
2016, 3, 19lspsncl 20824 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
2113, 18, 20syl2anc 583 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
223lsssubg 20804 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑦}) ∈ 𝑆) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
2313, 21, 22syl2anc 583 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
246adantr 480 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ 𝑆)
25 simprr 770 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ)
2616, 3lssel 20784 . . . . . . . . . . 11 ((π‘ˆ ∈ 𝑆 ∧ 𝑧 ∈ π‘ˆ) β†’ 𝑧 ∈ 𝑉)
2724, 25, 26syl2anc 583 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ 𝑉)
2816, 3, 19lspsncl 20824 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
2913, 27, 28syl2anc 583 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
303lsssubg 20804 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑧}) ∈ 𝑆) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3113, 29, 30syl2anc 583 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3216, 19lspsnid 20840 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3313, 18, 32syl2anc 583 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3416, 19lspsnid 20840 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
3513, 27, 34syl2anc 583 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
369, 10lsmelvali 19570 . . . . . . . 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 836 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))
38 eleq1a 2822 . . . . . . 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 20931 . . . . . . . 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 20841 . . . . . 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 3199 . . . 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 480 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
473, 19, 13, 14, 15lspsnel5a 20843 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) βŠ† 𝑇)
4810lsmless1 19580 . . . . . . . 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 480 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
513, 19, 13, 24, 25lspsnel5a 20843 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) βŠ† π‘ˆ)
5210lsmless2 19581 . . . . . . . 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 3987 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5554sseld 3976 . . . . 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 260 . . . 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 3205 . . 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 3184 . . . 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 3088 . . 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 3184 . . 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 275 . 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 287 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 395   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3064   βŠ† wss 3943  {csn 4623  β€˜cfv 6537  (class class class)co 7405  Basecbs 17153  +gcplusg 17206  SubGrpcsubg 19047  LSSumclsm 19554  LModclmod 20706  LSubSpclss 20778  LSpanclspn 20818
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  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 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-grp 18866  df-minusg 18867  df-sbg 18868  df-subg 19050  df-cntz 19233  df-lsm 19556  df-cmn 19702  df-abl 19703  df-mgp 20040  df-ur 20087  df-ring 20140  df-lmod 20708  df-lss 20779  df-lsp 20819
This theorem is referenced by:  dihjat1lem  40812
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