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Theorem f1lindf 20083
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 2621 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
21lindff 20076 . . . . . 6 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 469 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
433adant3 1079 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5 f1f 6060 . . . . 5 (𝐺:𝐾1-1→dom 𝐹𝐺:𝐾⟶dom 𝐹)
653ad2ant3 1082 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7 fco 6017 . . . 4 ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
84, 6, 7syl2anc 692 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
9 ffdm 6021 . . . 4 ((𝐹𝐺):𝐾⟶(Base‘𝑊) → ((𝐹𝐺):dom (𝐹𝐺)⟶(Base‘𝑊) ∧ dom (𝐹𝐺) ⊆ 𝐾))
109simpld 475 . . 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 1063 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ (𝑥 ∈ dom (𝐹𝐺) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}))) → 𝐹 LIndF 𝑊)
136adantr 481 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺:𝐾⟶dom 𝐹)
14 fdm 6010 . . . . . . . . . 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 2684 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝑥 ∈ dom (𝐹𝐺) ↔ 𝑥𝐾))
1716biimpa 501 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝑥𝐾)
1813, 17ffvelrnd 6318 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝐺𝑥) ∈ dom 𝐹)
1918adantrr 752 . . . . 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 3712 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
2120ad2antll 764 . . . . 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 4291 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
2322ad2antll 764 . . . . 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 2621 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
25 eqid 2621 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
26 eqid 2621 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
27 eqid 2621 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28 eqid 2621 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
2924, 25, 26, 27, 28lindfind 20077 . . . . 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 1324 . . . 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 6061 . . . . . . . . . . 11 (𝐺:𝐾1-1→dom 𝐹𝐺 Fn 𝐾)
32313ad2ant3 1082 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 Fn 𝐾)
3332adantr 481 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → 𝐺 Fn 𝐾)
34 fvco2 6232 . . . . . . . . 9 ((𝐺 Fn 𝐾𝑥𝐾) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3533, 17, 34syl2anc 692 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3635oveq2d 6623 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) = (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))))
3736eleq1d 2683 . . . . . 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 1062 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝑊 ∈ LMod)
39 imassrn 5438 . . . . . . . . . . 11 (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ ran 𝐹
40 frn 6012 . . . . . . . . . . . 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 3595 . . . . . . . . . 10 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
4342adantr 481 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ (Base‘𝑊))
44 imaco 5601 . . . . . . . . . 10 ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})))
4515difeq1d 3707 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (dom (𝐹𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
4645imaeq2d 5427 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
47 df-f1 5854 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun 𝐺))
4847simprbi 480 . . . . . . . . . . . . . . . 16 (𝐺:𝐾1-1→dom 𝐹 → Fun 𝐺)
49483ad2ant3 1082 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → Fun 𝐺)
50 imadif 5933 . . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . 13 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
5352adantr 481 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
54 fnsnfv 6217 . . . . . . . . . . . . . . 15 ((𝐺 Fn 𝐾𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5532, 54sylan 488 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → {(𝐺𝑥)} = (𝐺 “ {𝑥}))
5655difeq2d 3708 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
57 imassrn 5438 . . . . . . . . . . . . . . 15 (𝐺𝐾) ⊆ ran 𝐺
586adantr 481 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝐺:𝐾⟶dom 𝐹)
59 frn 6012 . . . . . . . . . . . . . . . 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 3595 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺𝐾) ⊆ dom 𝐹)
6261ssdifd 3726 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
6356, 62eqsstr3d 3621 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
6453, 63eqsstrd 3620 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
65 imass2 5462 . . . . . . . . . . 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 3630 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
681, 25lspss 18906 . . . . . . . . 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 1323 . . . . . . . 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 487 . . . . . . 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 3583 . . . . . 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 752 . . . 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 2965 . 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 1059 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝑊 ∈ LMod)
77 rellindf 20069 . . . . . 6 Rel LIndF
7877brrelexi 5120 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
79783ad2ant2 1081 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹 ∈ V)
80 simp3 1061 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾1-1→dom 𝐹)
81 dmexg 7047 . . . . . . 7 (𝐹 ∈ V → dom 𝐹 ∈ V)
8279, 81syl 17 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom 𝐹 ∈ V)
83 f1dmex 7086 . . . . . 6 ((𝐺:𝐾1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
8480, 82, 83syl2anc 692 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐾 ∈ V)
85 fex 6447 . . . . 5 ((𝐺:𝐾⟶dom 𝐹𝐾 ∈ V) → 𝐺 ∈ V)
866, 84, 85syl2anc 692 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 ∈ V)
87 coexg 7067 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝐺) ∈ V)
8879, 86, 87syl2anc 692 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) ∈ V)
891, 24, 25, 26, 28, 27islindf 20073 . . 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 692 . 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 956 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  Vcvv 3186  cdif 3553  wss 3556  {csn 4150   class class class wbr 4615  ccnv 5075  dom cdm 5076  ran crn 5077  cima 5079  ccom 5080  Fun wfun 5843   Fn wfn 5844  wf 5845  1-1wf1 5846  cfv 5849  (class class class)co 6607  Basecbs 15784  Scalarcsca 15868   ·𝑠 cvsca 15869  0gc0g 16024  LModclmod 18787  LSpanclspn 18893   LIndF clindf 20065
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 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905
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 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-slot 15788  df-base 15789  df-0g 16026  df-mgm 17166  df-sgrp 17208  df-mnd 17219  df-grp 17349  df-lmod 18789  df-lss 18855  df-lsp 18894  df-lindf 20067
This theorem is referenced by:  lindfres  20084  f1linds  20086
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