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Theorem f1lindf 20511
 Description: Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
f1lindf ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) LIndF 𝑊)

Proof of Theorem f1lindf
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2798 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
21lindff 20504 . . . . . 6 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 462 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
433adant3 1129 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5 f1f 6549 . . . . 5 (𝐺:𝐾1-1→dom 𝐹𝐺:𝐾⟶dom 𝐹)
653ad2ant3 1132 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7 fco 6505 . . . 4 ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
84, 6, 7syl2anc 587 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
98ffdmd 6511 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊))
10 simpl2 1189 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
116adantr 484 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺:𝐾⟶dom 𝐹)
128fdmd 6497 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom (𝐹𝐺) = 𝐾)
1312eleq2d 2875 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹𝐺) ↔ 𝑥𝐾))
1413biimpa 480 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝑥𝐾)
1511, 14ffvelrnd 6829 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝐺𝑥) ∈ dom 𝐹)
1615adantrr 716 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → (𝐺𝑥) ∈ dom 𝐹)
17 eldifi 4054 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
1817ad2antll 728 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
19 eldifsni 4683 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
2019ad2antll 728 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
21 eqid 2798 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
22 eqid 2798 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
23 eqid 2798 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
24 eqid 2798 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
25 eqid 2798 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
2621, 22, 23, 24, 25lindfind 20505 . . . . 5 (((𝐹 LIndF 𝑊 ∧ (𝐺𝑥) ∈ dom 𝐹) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑊)) ∧ 𝑘 ≠ (0g‘(Scalar‘𝑊)))) → ¬ (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))))
2710, 16, 18, 20, 26syl22anc 837 . . . 4 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))))
28 f1fn 6550 . . . . . . . . . . 11 (𝐺:𝐾1-1→dom 𝐹𝐺 Fn 𝐾)
29283ad2ant3 1132 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 Fn 𝐾)
3029adantr 484 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺 Fn 𝐾)
31 fvco2 6735 . . . . . . . . 9 ((𝐺 Fn 𝐾𝑥𝐾) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3230, 14, 31syl2anc 587 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3332oveq2d 7151 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) = (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))))
3433eleq1d 2874 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) ↔ (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})))))
35 simpl1 1188 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝑊 ∈ LMod)
36 imassrn 5907 . . . . . . . . . . 11 (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ ran 𝐹
374frnd 6494 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
3836, 37sstrid 3926 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
3938adantr 484 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
40 imaco 6071 . . . . . . . . . 10 ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})))
4112difeq1d 4049 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (dom (𝐹𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
4241imaeq2d 5896 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
43 df-f1 6329 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun 𝐺))
4443simprbi 500 . . . . . . . . . . . . . . . 16 (𝐺:𝐾1-1→dom 𝐹 → Fun 𝐺)
45443ad2ant3 1132 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → Fun 𝐺)
46 imadif 6408 . . . . . . . . . . . . . . 15 (Fun 𝐺 → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
4745, 46syl 17 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
4842, 47eqtrd 2833 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
4948adantr 484 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
50 fnsnfv 6718 . . . . . . . . . . . . . . 15 ((𝐺 Fn 𝐾𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5129, 50sylan 583 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5251difeq2d 4050 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
53 imassrn 5907 . . . . . . . . . . . . . . 15 (𝐺𝐾) ⊆ ran 𝐺
546adantr 484 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝐺:𝐾⟶dom 𝐹)
5554frnd 6494 . . . . . . . . . . . . . . 15 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ran 𝐺 ⊆ dom 𝐹)
5653, 55sstrid 3926 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺𝐾) ⊆ dom 𝐹)
5756ssdifd 4068 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
5852, 57eqsstrrd 3954 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
5949, 58eqsstrd 3953 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
60 imass2 5932 . . . . . . . . . . 11 ((𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}) → (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
6159, 60syl 17 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
6240, 61eqsstrid 3963 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
631, 22lspss 19749 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊) ∧ ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))) → ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))))
6435, 39, 62, 63syl3anc 1368 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))))
6514, 64syldan 594 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))))
6665sseld 3914 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) → (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))))
6734, 66sylbid 243 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) → (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))))
6867adantrr 716 . . . 4 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ((𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) → (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))) ∈ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))))
6927, 68mtod 201 . . 3 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → ¬ (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))))
7069ralrimivva 3156 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → ∀𝑥 ∈ dom (𝐹𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))))
71 simp1 1133 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝑊 ∈ LMod)
72 rellindf 20497 . . . . . 6 Rel LIndF
7372brrelex1i 5572 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
74733ad2ant2 1131 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹 ∈ V)
75 simp3 1135 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾1-1→dom 𝐹)
7674dmexd 7596 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom 𝐹 ∈ V)
77 f1dmex 7640 . . . . . 6 ((𝐺:𝐾1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
7875, 76, 77syl2anc 587 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐾 ∈ V)
79 fex 6966 . . . . 5 ((𝐺:𝐾⟶dom 𝐹𝐾 ∈ V) → 𝐺 ∈ V)
806, 78, 79syl2anc 587 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 ∈ V)
81 coexg 7616 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝐺) ∈ V)
8274, 80, 81syl2anc 587 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) ∈ V)
831, 21, 22, 23, 25, 24islindf 20501 . . 3 ((𝑊 ∈ LMod ∧ (𝐹𝐺) ∈ V) → ((𝐹𝐺) LIndF 𝑊 ↔ ((𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))))))
8471, 82, 83syl2anc 587 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → ((𝐹𝐺) LIndF 𝑊 ↔ ((𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊) ∧ ∀𝑥 ∈ dom (𝐹𝐺)∀𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) ¬ (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) ∈ ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))))))
859, 70, 84mpbir2and 712 1 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) LIndF 𝑊)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  Vcvv 3441   ∖ cdif 3878   ⊆ wss 3881  {csn 4525   class class class wbr 5030  ◡ccnv 5518  dom cdm 5519  ran crn 5520   “ cima 5522   ∘ ccom 5523  Fun wfun 6318   Fn wfn 6319  ⟶wf 6320  –1-1→wf1 6321  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  Scalarcsca 16560   ·𝑠 cvsca 16561  0gc0g 16705  LModclmod 19627  LSpanclspn 19736   LIndF clindf 20493 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 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-slot 16479  df-base 16481  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-lmod 19629  df-lss 19697  df-lsp 19737  df-lindf 20495 This theorem is referenced by:  lindfres  20512  f1linds  20514
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