MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1lindf Structured version   Visualization version   GIF version

Theorem f1lindf 19980
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 2610 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
21lindff 19973 . . . . . 6 ((𝐹 LIndF 𝑊𝑊 ∈ LMod) → 𝐹:dom 𝐹⟶(Base‘𝑊))
32ancoms 468 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹⟶(Base‘𝑊))
433adant3 1074 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹:dom 𝐹⟶(Base‘𝑊))
5 f1f 6014 . . . . 5 (𝐺:𝐾1-1→dom 𝐹𝐺:𝐾⟶dom 𝐹)
653ad2ant3 1077 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾⟶dom 𝐹)
7 fco 5971 . . . 4 ((𝐹:dom 𝐹⟶(Base‘𝑊) ∧ 𝐺:𝐾⟶dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
84, 6, 7syl2anc 691 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺):𝐾⟶(Base‘𝑊))
9 ffdm 5975 . . . 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 1058 . . . . 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 5964 . . . . . . . . . 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 2673 . . . . . . . 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 6268 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝐺𝑥) ∈ dom 𝐹)
1918adantrr 749 . . . . 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 3694 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ∈ (Base‘(Scalar‘𝑊)))
2120ad2antll 761 . . . . 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 4261 . . . . . 6 (𝑘 ∈ ((Base‘(Scalar‘𝑊)) ∖ {(0g‘(Scalar‘𝑊))}) → 𝑘 ≠ (0g‘(Scalar‘𝑊)))
2322ad2antll 761 . . . . 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 2610 . . . . . 6 ( ·𝑠𝑊) = ( ·𝑠𝑊)
25 eqid 2610 . . . . . 6 (LSpan‘𝑊) = (LSpan‘𝑊)
26 eqid 2610 . . . . . 6 (Scalar‘𝑊) = (Scalar‘𝑊)
27 eqid 2610 . . . . . 6 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
28 eqid 2610 . . . . . 6 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
2924, 25, 26, 27, 28lindfind 19974 . . . . 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 1319 . . . 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 6015 . . . . . . . . . . 11 (𝐺:𝐾1-1→dom 𝐹𝐺 Fn 𝐾)
32313ad2ant3 1077 . . . . . . . . . 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 6183 . . . . . . . . 9 ((𝐺 Fn 𝐾𝑥𝐾) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3533, 17, 34syl2anc 691 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
3635oveq2d 6565 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥 ∈ dom (𝐹𝐺)) → (𝑘( ·𝑠𝑊)((𝐹𝐺)‘𝑥)) = (𝑘( ·𝑠𝑊)(𝐹‘(𝐺𝑥))))
3736eleq1d 2672 . . . . . 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 1057 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝑊 ∈ LMod)
39 imassrn 5396 . . . . . . . . . . 11 (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})) ⊆ ran 𝐹
40 frn 5966 . . . . . . . . . . . 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 3579 . . . . . . . . . 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 5557 . . . . . . . . . 10 ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐹 “ (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})))
4515difeq1d 3689 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (dom (𝐹𝐺) ∖ {𝑥}) = (𝐾 ∖ {𝑥}))
4645imaeq2d 5385 . . . . . . . . . . . . . 14 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) = (𝐺 “ (𝐾 ∖ {𝑥})))
47 df-f1 5809 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾1-1→dom 𝐹 ↔ (𝐺:𝐾⟶dom 𝐹 ∧ Fun 𝐺))
4847simprbi 479 . . . . . . . . . . . . . . . 16 (𝐺:𝐾1-1→dom 𝐹 → Fun 𝐺)
49483ad2ant3 1077 . . . . . . . . . . . . . . 15 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → Fun 𝐺)
50 imadif 5887 . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . 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 6168 . . . . . . . . . . . . . . 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 3690 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) = ((𝐺𝐾) ∖ (𝐺 “ {𝑥})))
57 imassrn 5396 . . . . . . . . . . . . . . 15 (𝐺𝐾) ⊆ ran 𝐺
586adantr 480 . . . . . . . . . . . . . . . 16 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → 𝐺:𝐾⟶dom 𝐹)
59 frn 5966 . . . . . . . . . . . . . . . 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 3579 . . . . . . . . . . . . . 14 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺𝐾) ⊆ dom 𝐹)
6261ssdifd 3708 . . . . . . . . . . . . 13 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ {(𝐺𝑥)}) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
6356, 62eqsstr3d 3603 . . . . . . . . . . . 12 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐺𝐾) ∖ (𝐺 “ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
6453, 63eqsstrd 3602 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → (𝐺 “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (dom 𝐹 ∖ {(𝐺𝑥)}))
65 imass2 5420 . . . . . . . . . . 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 3612 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) ∧ 𝑥𝐾) → ((𝐹𝐺) “ (dom (𝐹𝐺) ∖ {𝑥})) ⊆ (𝐹 “ (dom 𝐹 ∖ {(𝐺𝑥)})))
681, 25lspss 18805 . . . . . . . . 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 1318 . . . . . . . 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 3567 . . . . . 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 229 . . . . 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 749 . . . 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 188 . . 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 2954 . 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 1054 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝑊 ∈ LMod)
77 rellindf 19966 . . . . . 6 Rel LIndF
7877brrelexi 5082 . . . . 5 (𝐹 LIndF 𝑊𝐹 ∈ V)
79783ad2ant2 1076 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐹 ∈ V)
80 simp3 1056 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺:𝐾1-1→dom 𝐹)
81 dmexg 6989 . . . . . . 7 (𝐹 ∈ V → dom 𝐹 ∈ V)
8279, 81syl 17 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → dom 𝐹 ∈ V)
83 f1dmex 7029 . . . . . 6 ((𝐺:𝐾1-1→dom 𝐹 ∧ dom 𝐹 ∈ V) → 𝐾 ∈ V)
8480, 82, 83syl2anc 691 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐾 ∈ V)
85 fex 6394 . . . . 5 ((𝐺:𝐾⟶dom 𝐹𝐾 ∈ V) → 𝐺 ∈ V)
866, 84, 85syl2anc 691 . . . 4 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → 𝐺 ∈ V)
87 coexg 7010 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (𝐹𝐺) ∈ V)
8879, 86, 87syl2anc 691 . . 3 ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) ∈ V)
891, 24, 25, 26, 28, 27islindf 19970 . . 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 691 . 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 959 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wral 2896  Vcvv 3173  cdif 3537  wss 3540  {csn 4125   class class class wbr 4583  ccnv 5037  dom cdm 5038  ran crn 5039  cima 5041  ccom 5042  Fun wfun 5798   Fn wfn 5799  wf 5800  1-1wf1 5801  cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LModclmod 18686  LSpanclspn 18792   LIndF clindf 19962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-slot 15699  df-base 15700  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lindf 19964
This theorem is referenced by:  lindfres  19981  f1linds  19983
  Copyright terms: Public domain W3C validator