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Theorem f1lindf 21773
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 2725 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
21lindff 21766 . . . . . 6 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 457 . . . . 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 6793 . . . . 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 6747 . . . 4 ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
84, 6, 7syl2anc 582 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
98ffdmd 6754 . 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 479 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺:𝐾⟶dom 𝐹)
128fdmd 6733 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom (𝐹𝐺) = 𝐾)
1312eleq2d 2811 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹𝐺) ↔ 𝑥𝐾))
1413biimpa 475 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝑥𝐾)
1511, 14ffvelcdmd 7094 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝐺𝑥) ∈ dom 𝐹)
1615adantrr 715 . . . . 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 4123 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
1817ad2antll 727 . . . . 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 4795 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
2019ad2antll 727 . . . . 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 2725 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
22 eqid 2725 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
23 eqid 2725 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
24 eqid 2725 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
25 eqid 2725 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
2621, 22, 23, 24, 25lindfind 21767 . . . . 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 6794 . . . . . . . . . . 11 (𝐺:𝐾1-1→dom 𝐹𝐺 Fn 𝐾)
29283ad2ant3 1132 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 Fn 𝐾)
3029adantr 479 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺 Fn 𝐾)
31 fvco2 6994 . . . . . . . . 9 ((𝐺 Fn 𝐾𝑥𝐾) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3230, 14, 31syl2anc 582 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3332oveq2d 7435 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) = (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))))
3433eleq1d 2810 . . . . . 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 6075 . . . . . . . . . . 11 (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ ran 𝐹
374frnd 6731 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
3836, 37sstrid 3988 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
3938adantr 479 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
40 imaco 6257 . . . . . . . . . 10 ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})))
4112difeq1d 4117 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (dom (𝐹𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
4241imaeq2d 6064 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
43 df-f1 6554 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun 𝐺))
4443simprbi 495 . . . . . . . . . . . . . . . 16 (𝐺:𝐾1-1→dom 𝐹 → Fun 𝐺)
45443ad2ant3 1132 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → Fun 𝐺)
46 imadif 6638 . . . . . . . . . . . . . . 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 2765 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
4948adantr 479 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
50 fnsnfv 6976 . . . . . . . . . . . . . . 15 ((𝐺 Fn 𝐾𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5129, 50sylan 578 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5251difeq2d 4118 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
53 imassrn 6075 . . . . . . . . . . . . . . 15 (𝐺𝐾) ⊆ ran 𝐺
546adantr 479 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝐺:𝐾⟶dom 𝐹)
5554frnd 6731 . . . . . . . . . . . . . . 15 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ran 𝐺 ⊆ dom 𝐹)
5653, 55sstrid 3988 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺𝐾) ⊆ dom 𝐹)
5756ssdifd 4137 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
5852, 57eqsstrrd 4016 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
5949, 58eqsstrd 4015 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
60 imass2 6107 . . . . . . . . . . 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 4025 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
631, 22lspss 20880 . . . . . . . . 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 589 . . . . . . 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 3975 . . . . . 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 239 . . . . 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 715 . . . 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 197 . . 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 3190 . 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 21759 . . . . . 6 Rel LIndF
7372brrelex1i 5734 . . . . 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 7911 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom 𝐹 ∈ V)
77 f1dmex 7961 . . . . . 6 ((𝐺:𝐾1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
7875, 76, 77syl2anc 582 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐾 ∈ V)
796, 78fexd 7239 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 ∈ V)
80 coexg 7937 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝐺) ∈ V)
8174, 79, 80syl2anc 582 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) ∈ V)
821, 21, 22, 23, 25, 24islindf 21763 . . 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}))))))
8371, 81, 82syl2anc 582 . 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}))))))
849, 70, 83mpbir2and 711 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 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wral 3050  Vcvv 3461  cdif 3941  wss 3944  {csn 4630   class class class wbr 5149  ccnv 5677  dom cdm 5678  ran crn 5679  cima 5681  ccom 5682  Fun wfun 6543   Fn wfn 6544  wf 6545  1-1wf1 6546  cfv 6549  (class class class)co 7419  Basecbs 17183  Scalarcsca 17239   ·𝑠 cvsca 17240  0gc0g 17424  LModclmod 20755  LSpanclspn 20867   LIndF clindf 21755
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-1cn 11198  ax-addcl 11200
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-om 7872  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-nn 12246  df-slot 17154  df-ndx 17166  df-base 17184  df-0g 17426  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-grp 18901  df-lmod 20757  df-lss 20828  df-lsp 20868  df-lindf 21757
This theorem is referenced by:  lindfres  21774  f1linds  21776
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