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Theorem lindfrn 20079
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 2621 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
21lindff 20073 . . . 4 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 469 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
4 frn 6010 . . 3 (𝐹:dom 𝐹⟶(Base‘𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
53, 4syl 17 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
6 simpll 789 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → 𝑊 ∈ LMod)
7 imassrn 5436 . . . . . . . . 9 (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ ran 𝐹
87, 5syl5ss 3594 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
98adantr 481 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {𝑦})) ⊆ (Base‘𝑊))
10 ffun 6005 . . . . . . . . 9 (𝐹:dom 𝐹⟶(Base‘𝑊) → Fun 𝐹)
113, 10syl 17 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → Fun 𝐹)
12 eldifsn 4287 . . . . . . . . . 10 (𝑥 ∈ (ran 𝐹 ∖ {(𝐹𝑦)}) ↔ (𝑥 ∈ ran 𝐹𝑥 ≠ (𝐹𝑦)))
13 funfn 5877 . . . . . . . . . . . . . 14 (Fun 𝐹𝐹 Fn dom 𝐹)
14 fvelrnb 6200 . . . . . . . . . . . . . 14 (𝐹 Fn dom 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥))
1513, 14sylbi 207 . . . . . . . . . . . . 13 (Fun 𝐹 → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥))
1615adantr 481 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 ↔ ∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥))
17 difss 3715 . . . . . . . . . . . . . . . . . 18 (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹
1817jctr 564 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
1918ad2antrr 761 . . . . . . . . . . . . . . . 16 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦))) → (Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹))
20 simpl 473 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦)) → 𝑘 ∈ dom 𝐹)
21 fveq2 6148 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
2221necon3i 2822 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑘) ≠ (𝐹𝑦) → 𝑘𝑦)
2322adantl 482 . . . . . . . . . . . . . . . . . 18 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦)) → 𝑘𝑦)
24 eldifsn 4287 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) ↔ (𝑘 ∈ dom 𝐹𝑘𝑦))
2520, 23, 24sylanbrc 697 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦)) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
2625adantl 482 . . . . . . . . . . . . . . . 16 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦))) → 𝑘 ∈ (dom 𝐹 ∖ {𝑦}))
27 funfvima2 6447 . . . . . . . . . . . . . . . 16 ((Fun 𝐹 ∧ (dom 𝐹 ∖ {𝑦}) ⊆ dom 𝐹) → (𝑘 ∈ (dom 𝐹 ∖ {𝑦}) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
2819, 26, 27sylc 65 . . . . . . . . . . . . . . 15 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ≠ (𝐹𝑦))) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
2928expr 642 . . . . . . . . . . . . . 14 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹𝑘) ≠ (𝐹𝑦) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
30 neeq1 2852 . . . . . . . . . . . . . . 15 ((𝐹𝑘) = 𝑥 → ((𝐹𝑘) ≠ (𝐹𝑦) ↔ 𝑥 ≠ (𝐹𝑦)))
31 eleq1 2686 . . . . . . . . . . . . . . 15 ((𝐹𝑘) = 𝑥 → ((𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})) ↔ 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
3230, 31imbi12d 334 . . . . . . . . . . . . . 14 ((𝐹𝑘) = 𝑥 → (((𝐹𝑘) ≠ (𝐹𝑦) → (𝐹𝑘) ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))) ↔ (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3329, 32syl5ibcom 235 . . . . . . . . . . . . 13 (((Fun 𝐹𝑦 ∈ dom 𝐹) ∧ 𝑘 ∈ dom 𝐹) → ((𝐹𝑘) = 𝑥 → (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3433rexlimdva 3024 . . . . . . . . . . . 12 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (∃𝑘 ∈ dom 𝐹(𝐹𝑘) = 𝑥 → (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3516, 34sylbid 230 . . . . . . . . . . 11 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑥 ∈ ran 𝐹 → (𝑥 ≠ (𝐹𝑦) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦})))))
3635impd 447 . . . . . . . . . 10 ((Fun 𝐹𝑦 ∈ dom 𝐹) → ((𝑥 ∈ ran 𝐹𝑥 ≠ (𝐹𝑦)) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
3712, 36syl5bi 232 . . . . . . . . 9 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝑥 ∈ (ran 𝐹 ∖ {(𝐹𝑦)}) → 𝑥 ∈ (𝐹 “ (dom 𝐹 ∖ {𝑦}))))
3837ssrdv 3589 . . . . . . . 8 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
3911, 38sylan 488 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → (ran 𝐹 ∖ {(𝐹𝑦)}) ⊆ (𝐹 “ (dom 𝐹 ∖ {𝑦})))
40 eqid 2621 . . . . . . . 8 (LSpan‘𝑊) = (LSpan‘𝑊)
411, 40lspss 18903 . . . . . . 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}))))
426, 9, 39, 41syl3anc 1323 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ 𝑦 ∈ dom 𝐹) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
4342adantrr 752 . . . . 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}))))
44 simplr 791 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
45 simprl 793 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑦 ∈ dom 𝐹)
46 eldifi 3710 . . . . . . 7 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
4746ad2antll 764 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
48 eldifsni 4289 . . . . . . 7 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
4948ad2antll 764 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
50 eqid 2621 . . . . . . 7 ( ·𝑠𝑊) = ( ·𝑠𝑊)
51 eqid 2621 . . . . . . 7 (Scalar‘𝑊) = (Scalar‘𝑊)
52 eqid 2621 . . . . . . 7 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
53 eqid 2621 . . . . . . 7 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
5450, 40, 51, 52, 53lindfind 20074 . . . . . 6 (((𝐹 LIndF 𝑊𝑦 ∈ dom 𝐹) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊)))) → ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
5544, 45, 47, 49, 54syl22anc 1324 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {𝑦}))))
5643, 55ssneldd 3586 . . . 4 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) ∧ (𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})))
5756ralrimivva 2965 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ∀𝑦 ∈ dom 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})))
5811, 13sylib 208 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹 Fn dom 𝐹)
59 oveq2 6612 . . . . . . . 8 (𝑥 = (𝐹𝑦) → (𝑘( ·𝑠𝑊)𝑥) = (𝑘( ·𝑠𝑊)(𝐹𝑦)))
60 sneq 4158 . . . . . . . . . 10 (𝑥 = (𝐹𝑦) → {𝑥} = {(𝐹𝑦)})
6160difeq2d 3706 . . . . . . . . 9 (𝑥 = (𝐹𝑦) → (ran 𝐹 ∖ {𝑥}) = (ran 𝐹 ∖ {(𝐹𝑦)}))
6261fveq2d 6152 . . . . . . . 8 (𝑥 = (𝐹𝑦) → ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) = ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)})))
6359, 62eleq12d 2692 . . . . . . 7 (𝑥 = (𝐹𝑦) → ((𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6463notbid 308 . . . . . 6 (𝑥 = (𝐹𝑦) → (¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})) ↔ ¬ (𝑘( ·𝑠𝑊)(𝐹𝑦)) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {(𝐹𝑦)}))))
6564ralbidv 2980 . . . . 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)}))))
6665ralrn 6318 . . . 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)}))))
6758, 66syl 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)}))))
6857, 67mpbird 247 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ∀𝑥 ∈ ran 𝐹𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)𝑥) ∈ ((LSpan‘𝑊)‘(ran 𝐹 ∖ {𝑥})))
691, 50, 40, 51, 53, 52islinds2 20071 . . 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})))))
7069adantr 481 . 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})))))
715, 68, 70mpbir2and 956 1 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  cdif 3552  wss 3555  {csn 4148   class class class wbr 4613  dom cdm 5074  ran crn 5075  cima 5077  Fun wfun 5841   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  Basecbs 15781  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  LModclmod 18784  LSpanclspn 18890   LIndF clindf 20062  LIndSclinds 20063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-slot 15785  df-base 15786  df-0g 16023  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-grp 17346  df-lmod 18786  df-lss 18852  df-lsp 18891  df-lindf 20064  df-linds 20065
This theorem is referenced by:  islindf3  20084  lindsmm  20086  matunitlindflem2  33035
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