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Theorem lindfrn 20599
 Description: The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
lindfrn ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))

Proof of Theorem lindfrn
Dummy variables 𝑘 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2758 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
21lindff 20593 . . . 4 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 462 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
43frnd 6510 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
5 simpll 766 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → 𝑊 ∈ LMod)
6 imassrn 5917 . . . . . . . . 9 (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ ran 𝐹
76, 4sstrid 3905 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
87adantr 484 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
93ffund 6507 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → Fun 𝐹)
10 eldifsn 4680 . . . . . . . . . 10 (𝑥 ∈ (ran 𝐹 ∖ {(𝐹𝑦)}) ↔ (𝑥 ∈ ran 𝐹𝑥 ≠ (𝐹𝑦)))
11 funfn 6370 . . . . . . . . . . . . . 14 (Fun 𝐹𝐹 Fn dom 𝐹)
12 fvelrnb 6719 . . . . . . . . . . . . . 14 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥))
1311, 12sylbi 220 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥))
1413adantr 484 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥))
15 difss 4039 . . . . . . . . . . . . . . . . . 18 (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹
1615jctr 528 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
1716ad2antrr 725 . . . . . . . . . . . . . . . 16 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦))) → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
18 simpl 486 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦)) → 𝑘 ∈ dom 𝐹)
19 fveq2 6663 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
2019necon3i 2983 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑘) ≠ (𝐹𝑦) → 𝑘𝑦)
2120adantl 485 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦)) → 𝑘𝑦)
22 eldifsn 4680 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) ↔ (𝑘 ∈ dom 𝐹𝑘𝑦))
2318, 21, 22sylanbrc 586 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦)) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
2423adantl 485 . . . . . . . . . . . . . . . 16 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦))) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
25 funfvima2 6991 . . . . . . . . . . . . . . . 16 ((Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹) → (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
2617, 24, 25sylc 65 . . . . . . . . . . . . . . 15 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦))) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
2726expr 460 . . . . . . . . . . . . . 14 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹𝑘) ≠ (𝐹𝑦) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
28 neeq1 3013 . . . . . . . . . . . . . . 15 ((𝐹𝑘) = 𝑥 → ((𝐹𝑘) ≠ (𝐹𝑦) ↔ 𝑥 ≠ (𝐹𝑦)))
29 eleq1 2839 . . . . . . . . . . . . . . 15 ((𝐹𝑘) = 𝑥 → ((𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ↔ 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
3028, 29imbi12d 348 . . . . . . . . . . . . . 14 ((𝐹𝑘) = 𝑥 → (((𝐹𝑘) ≠ (𝐹𝑦) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))) ↔ (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3127, 30syl5ibcom 248 . . . . . . . . . . . . 13 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹𝑘) = 𝑥 → (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3231rexlimdva 3208 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥 → (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3314, 32sylbid 243 . . . . . . . . . . 11 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3433impd 414 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ dom 𝐹) → ((𝑥 ∈ ran 𝐹𝑥 ≠ (𝐹𝑦)) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
3510, 34syl5bi 245 . . . . . . . . 9 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑥 ∈ (ran 𝐹 ∖ {(𝐹𝑦)}) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
3635ssrdv 3900 . . . . . . . 8 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
379, 36sylan 583 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
38 eqid 2758 . . . . . . . 8 (LSpan‘𝑊) = (LSpan‘𝑊)
391, 38lspss 19837 . . . . . . 7 ((𝑊 ∈ LMod ∧ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊) ∧ (ran 𝐹 ∖ {(𝐹𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦}))) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
405, 8, 37, 39syl3anc 1368 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
4140adantrr 716 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
42 simplr 768 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
43 simprl 770 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑦 ∈ dom 𝐹)
44 eldifi 4034 . . . . . . 7 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
4544ad2antll 728 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
46 eldifsni 4683 . . . . . . 7 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
4746ad2antll 728 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
48 eqid 2758 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
49 eqid 2758 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
50 eqid 2758 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
51 eqid 2758 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
5248, 38, 49, 50, 51lindfind 20594 . . . . . 6 (((𝐹 LIndF 𝑊𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊)))) → ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
5342, 43, 45, 47, 52syl22anc 837 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
5441, 53ssneldd 3897 . . . 4 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})))
5554ralrimivva 3120 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ∀𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})))
569funfnd 6371 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹 Fn dom 𝐹)
57 oveq2 7164 . . . . . . . 8 (𝑥 = (𝐹𝑦) → (𝑘( ·𝑠𝑊)𝑥) = (𝑘( ·𝑠𝑊)(𝐹𝑦)))
58 sneq 4535 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → {𝑥} = {(𝐹𝑦)})
5958difeq2d 4030 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (ran 𝐹 ∖ {𝑥}) = (ran 𝐹 ∖ {(𝐹𝑦)}))
6059fveq2d 6667 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) = ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})))
6157, 60eleq12d 2846 . . . . . . 7 (𝑥 = (𝐹𝑦) → ((𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6261notbid 321 . . . . . 6 (𝑥 = (𝐹𝑦) → (¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6362ralbidv 3126 . . . . 5 (𝑥 = (𝐹𝑦) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6463ralrn 6851 . . . 4 (𝐹 Fn dom 𝐹 → (∀𝑥 ∈ ran 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ∀𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6556, 64syl 17 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (∀𝑥 ∈ ran 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ∀𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6655, 65mpbird 260 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ∀𝑥 ∈ ran 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))
671, 48, 38, 49, 51, 50islinds2 20591 . . 3 (𝑊 ∈ LMod → (ran 𝐹 ∈ (LIndS‘𝑊) ↔ (ran 𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ran 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))))
6867adantr 484 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (ran 𝐹 ∈ (LIndS‘𝑊) ↔ (ran 𝐹 ⊆ (Base‘𝑊) ∧ ∀𝑥 ∈ ran 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))))
694, 66, 68mpbir2and 712 1 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2951  ∀wral 3070  ∃wrex 3071   ∖ cdif 3857   ⊆ wss 3860  {csn 4525   class class class wbr 5036  dom cdm 5528  ran crn 5529   “ cima 5531  Fun wfun 6334   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  Scalarcsca 16639   ·𝑠 cvsca 16640  0gc0g 16784  LModclmod 19715  LSpanclspn 19824   LIndF clindf 20582  LIndSclinds 20583 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-slot 16558  df-base 16560  df-0g 16786  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-grp 18185  df-lmod 19717  df-lss 19785  df-lsp 19825  df-lindf 20584  df-linds 20585 This theorem is referenced by:  islindf3  20604  lindsmm  20606  matunitlindflem2  35368
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