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Theorem lsmelval2 20561
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 20433 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝑆) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
51, 2, 4syl2anc 585 . . . . 5 (πœ‘ β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
6 lsmelval2.u . . . . . 6 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
73lsssubg 20433 . . . . . 6 ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
81, 6, 7syl2anc 585 . . . . 5 (πœ‘ β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
9 eqid 2733 . . . . . 6 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
10 lsmelval2.p . . . . . 6 βŠ• = (LSSumβ€˜π‘Š)
119, 10lsmelval 19436 . . . . 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 585 . . . 4 (πœ‘ β†’ (𝑋 ∈ (𝑇 βŠ• π‘ˆ) ↔ βˆƒπ‘¦ ∈ 𝑇 βˆƒπ‘§ ∈ π‘ˆ 𝑋 = (𝑦(+gβ€˜π‘Š)𝑧)))
131adantr 482 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘Š ∈ LMod)
142adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ 𝑆)
15 simprl 770 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑇)
16 lsmelval2.v . . . . . . . . . . . 12 𝑉 = (Baseβ€˜π‘Š)
1716, 3lssel 20413 . . . . . . . . . . 11 ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) β†’ 𝑦 ∈ 𝑉)
1814, 15, 17syl2anc 585 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ 𝑉)
19 lsmelval2.n . . . . . . . . . . 11 𝑁 = (LSpanβ€˜π‘Š)
2016, 3, 19lspsncl 20453 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
2113, 18, 20syl2anc 585 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ 𝑆)
223lsssubg 20433 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑦}) ∈ 𝑆) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
2313, 21, 22syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) ∈ (SubGrpβ€˜π‘Š))
246adantr 482 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ 𝑆)
25 simprr 772 . . . . . . . . . . 11 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ π‘ˆ)
2616, 3lssel 20413 . . . . . . . . . . 11 ((π‘ˆ ∈ 𝑆 ∧ 𝑧 ∈ π‘ˆ) β†’ 𝑧 ∈ 𝑉)
2724, 25, 26syl2anc 585 . . . . . . . . . 10 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ 𝑉)
2816, 3, 19lspsncl 20453 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
2913, 27, 28syl2anc 585 . . . . . . . . 9 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ 𝑆)
303lsssubg 20433 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (π‘β€˜{𝑧}) ∈ 𝑆) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3113, 29, 30syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) ∈ (SubGrpβ€˜π‘Š))
3216, 19lspsnid 20469 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑦 ∈ 𝑉) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3313, 18, 32syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑦 ∈ (π‘β€˜{𝑦}))
3416, 19lspsnid 20469 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝑧 ∈ 𝑉) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
3513, 27, 34syl2anc 585 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑧 ∈ (π‘β€˜{𝑧}))
369, 10lsmelvali 19437 . . . . . . . 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 838 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑦(+gβ€˜π‘Š)𝑧) ∈ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})))
38 eleq1a 2829 . . . . . . 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 20559 . . . . . . . 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 1372 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) ∈ 𝑆)
4216, 3, 19, 13, 41lspsnel6 20470 . . . . . 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 482 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ 𝑇 ∈ (SubGrpβ€˜π‘Š))
473, 19, 13, 14, 15lspsnel5a 20472 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑦}) βŠ† 𝑇)
4810lsmless1 19447 . . . . . . . 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 1372 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• (π‘β€˜{𝑧})))
508adantr 482 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
513, 19, 13, 24, 25lspsnel5a 20472 . . . . . . . 8 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (π‘β€˜{𝑧}) βŠ† π‘ˆ)
5210lsmless2 19448 . . . . . . . 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 1372 . . . . . . 7 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ (𝑇 βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5449, 53sstrd 3955 . . . . . 6 ((πœ‘ ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ π‘ˆ)) β†’ ((π‘β€˜{𝑦}) βŠ• (π‘β€˜{𝑧})) βŠ† (𝑇 βŠ• π‘ˆ))
5554sseld 3944 . . . . 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 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 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 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 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   βŠ† wss 3911  {csn 4587  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  SubGrpcsubg 18927  LSSumclsm 19421  LModclmod 20336  LSubSpclss 20407  LSpanclspn 20447
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 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-nn 12159  df-2 12221  df-sets 17041  df-slot 17059  df-ndx 17071  df-base 17089  df-ress 17118  df-plusg 17151  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-submnd 18607  df-grp 18756  df-minusg 18757  df-sbg 18758  df-subg 18930  df-cntz 19102  df-lsm 19423  df-cmn 19569  df-abl 19570  df-mgp 19902  df-ur 19919  df-ring 19971  df-lmod 20338  df-lss 20408  df-lsp 20448
This theorem is referenced by:  dihjat1lem  39937
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