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Theorem f1lindf 20209
 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 2651 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
21lindff 20202 . . . . . 6 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 468 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
433adant3 1101 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5 f1f 6139 . . . . 5 (𝐺:𝐾1-1→dom 𝐹𝐺:𝐾⟶dom 𝐹)
653ad2ant3 1104 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7 fco 6096 . . . 4 ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
84, 6, 7syl2anc 694 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
9 ffdm 6100 . . . 4 ((𝐹𝐺):𝐾⟶(Base‘𝑊) → ((𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊) ∧ dom (𝐹𝐺) ⊆ 𝐾))
109simpld 474 . . 3 ((𝐹𝐺):𝐾⟶(Base‘𝑊) → (𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊))
118, 10syl 17 . 2 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊))
12 simpl2 1085 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
136adantr 480 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺:𝐾⟶dom 𝐹)
14 fdm 6089 . . . . . . . . . 10 ((𝐹𝐺):𝐾⟶(Base‘𝑊) → dom (𝐹𝐺) = 𝐾)
158, 14syl 17 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom (𝐹𝐺) = 𝐾)
1615eleq2d 2716 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹𝐺) ↔ 𝑥𝐾))
1716biimpa 500 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝑥𝐾)
1813, 17ffvelrnd 6400 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝐺𝑥) ∈ dom 𝐹)
1918adantrr 753 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → (𝐺𝑥) ∈ dom 𝐹)
20 eldifi 3765 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
2120ad2antll 765 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
22 eldifsni 4353 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
2322ad2antll 765 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
24 eqid 2651 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
25 eqid 2651 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
26 eqid 2651 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
27 eqid 2651 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28 eqid 2651 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
2924, 25, 26, 27, 28lindfind 20203 . . . . 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)}))))
3012, 19, 21, 23, 29syl22anc 1367 . . . 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)}))))
31 f1fn 6140 . . . . . . . . . . 11 (𝐺:𝐾1-1→dom 𝐹𝐺 Fn 𝐾)
32313ad2ant3 1104 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 Fn 𝐾)
3332adantr 480 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺 Fn 𝐾)
34 fvco2 6312 . . . . . . . . 9 ((𝐺 Fn 𝐾𝑥𝐾) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3533, 17, 34syl2anc 694 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3635oveq2d 6706 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) = (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))))
3736eleq1d 2715 . . . . . 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})))))
38 simpl1 1084 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝑊 ∈ LMod)
39 imassrn 5512 . . . . . . . . . . 11 (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ ran 𝐹
40 frn 6091 . . . . . . . . . . . 12 (𝐹:dom 𝐹⟶(Base‘𝑊) → ran 𝐹 ⊆ (Base‘𝑊))
414, 40syl 17 . . . . . . . . . . 11 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → ran 𝐹 ⊆ (Base‘𝑊))
4239, 41syl5ss 3647 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
4342adantr 480 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
44 imaco 5678 . . . . . . . . . 10 ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})))
4515difeq1d 3760 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (dom (𝐹𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
4645imaeq2d 5501 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
47 df-f1 5931 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun 𝐺))
4847simprbi 479 . . . . . . . . . . . . . . . 16 (𝐺:𝐾1-1→dom 𝐹 → Fun 𝐺)
49483ad2ant3 1104 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → Fun 𝐺)
50 imadif 6011 . . . . . . . . . . . . . . 15 (Fun 𝐺 → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
5149, 50syl 17 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (𝐾 ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
5246, 51eqtrd 2685 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
5352adantr 480 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
54 fnsnfv 6297 . . . . . . . . . . . . . . 15 ((𝐺 Fn 𝐾𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5532, 54sylan 487 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5655difeq2d 3761 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
57 imassrn 5512 . . . . . . . . . . . . . . 15 (𝐺𝐾) ⊆ ran 𝐺
586adantr 480 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝐺:𝐾⟶dom 𝐹)
59 frn 6091 . . . . . . . . . . . . . . . 16 (𝐺:𝐾⟶dom 𝐹 → ran 𝐺 ⊆ dom 𝐹)
6058, 59syl 17 . . . . . . . . . . . . . . 15 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ran 𝐺 ⊆ dom 𝐹)
6157, 60syl5ss 3647 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺𝐾) ⊆ dom 𝐹)
6261ssdifd 3779 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
6356, 62eqsstr3d 3673 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
6453, 63eqsstrd 3672 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
65 imass2 5536 . . . . . . . . . . 11 ((𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}) → (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
6664, 65syl 17 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
6744, 66syl5eqss 3682 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
681, 25lspss 19032 . . . . . . . . 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)}))))
6938, 43, 67, 68syl3anc 1366 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((LSpan‘𝑊)‘((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥}))) ⊆ ((LSpan‘𝑊)‘(𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)}))))
7017, 69syldan 486 . . . . . . 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)}))))
7170sseld 3635 . . . . . 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)})))))
7237, 71sylbid 230 . . . . 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)})))))
7372adantrr 753 . . . 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)})))))
7430, 73mtod 189 . . 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}))))
7574ralrimivva 3000 . 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}))))
76 simp1 1081 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝑊 ∈ LMod)
77 rellindf 20195 . . . . . 6 Rel LIndF
7877brrelexi 5192 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
79783ad2ant2 1103 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹 ∈ V)
80 simp3 1083 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾1-1→dom 𝐹)
81 dmexg 7139 . . . . . . 7 (𝐹 ∈ V → dom 𝐹 ∈ V)
8279, 81syl 17 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom 𝐹 ∈ V)
83 f1dmex 7178 . . . . . 6 ((𝐺:𝐾1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
8480, 82, 83syl2anc 694 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐾 ∈ V)
85 fex 6530 . . . . 5 ((𝐺:𝐾⟶dom 𝐹𝐾 ∈ V) → 𝐺 ∈ V)
866, 84, 85syl2anc 694 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 ∈ V)
87 coexg 7159 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝐺) ∈ V)
8879, 86, 87syl2anc 694 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) ∈ V)
891, 24, 25, 26, 28, 27islindf 20199 . . 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}))))))
9076, 88, 89syl2anc 694 . 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}))))))
9111, 75, 90mpbir2and 977 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  {csn 4210   class class class wbr 4685  ◡ccnv 5142  dom cdm 5143  ran crn 5144   “ cima 5146   ∘ ccom 5147  Fun wfun 5920   Fn wfn 5921  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  (class class class)co 6690  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  LModclmod 18911  LSpanclspn 19019   LIndF clindf 20191 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-slot 15908  df-base 15910  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lindf 20193 This theorem is referenced by:  lindfres  20210  f1linds  20212
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