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Theorem f1lindf 21377
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 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺) LIndF π‘Š)
Proof of Theorem f1lindf
Dummy variables π‘˜ π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
21lindff 21370 . . . . . 6 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ LMod) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
32ancoms 460 . . . . 5 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
433adant3 1133 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
5 f1f 6788 . . . . 5 (𝐺:𝐾–1-1β†’dom 𝐹 β†’ 𝐺:𝐾⟢dom 𝐹)
653ad2ant3 1136 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺:𝐾⟢dom 𝐹)
7 fco 6742 . . . 4 ((𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) ∧ 𝐺:𝐾⟢dom 𝐹) β†’ (𝐹 ∘ 𝐺):𝐾⟢(Baseβ€˜π‘Š))
84, 6, 7syl2anc 585 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺):𝐾⟢(Baseβ€˜π‘Š))
98ffdmd 6749 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟢(Baseβ€˜π‘Š))
10 simpl2 1193 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝐹 LIndF π‘Š)
116adantr 482 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ 𝐺:𝐾⟢dom 𝐹)
128fdmd 6729 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ dom (𝐹 ∘ 𝐺) = 𝐾)
1312eleq2d 2820 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ↔ π‘₯ ∈ 𝐾))
1413biimpa 478 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ π‘₯ ∈ 𝐾)
1511, 14ffvelcdmd 7088 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ (πΊβ€˜π‘₯) ∈ dom 𝐹)
1615adantrr 716 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ (πΊβ€˜π‘₯) ∈ dom 𝐹)
17 eldifi 4127 . . . . . 6 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
1817ad2antll 728 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
19 eldifsni 4794 . . . . . 6 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
2019ad2antll 728 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
21 eqid 2733 . . . . . 6 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
22 eqid 2733 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
23 eqid 2733 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
24 eqid 2733 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
25 eqid 2733 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
2621, 22, 23, 24, 25lindfind 21371 . . . . 5 (((𝐹 LIndF π‘Š ∧ (πΊβ€˜π‘₯) ∈ dom 𝐹) ∧ (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))))
2710, 16, 18, 20, 26syl22anc 838 . . . 4 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))))
28 f1fn 6789 . . . . . . . . . . 11 (𝐺:𝐾–1-1β†’dom 𝐹 β†’ 𝐺 Fn 𝐾)
29283ad2ant3 1136 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺 Fn 𝐾)
3029adantr 482 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ 𝐺 Fn 𝐾)
31 fvco2 6989 . . . . . . . . 9 ((𝐺 Fn 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ ((𝐹 ∘ 𝐺)β€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘₯)))
3230, 14, 31syl2anc 585 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ ((𝐹 ∘ 𝐺)β€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘₯)))
3332oveq2d 7425 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ (π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) = (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))))
3433eleq1d 2819 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) ↔ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})))))
35 simpl1 1192 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ π‘Š ∈ LMod)
36 imassrn 6071 . . . . . . . . . . 11 (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})) βŠ† ran 𝐹
374frnd 6726 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘Š))
3836, 37sstrid 3994 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})) βŠ† (Baseβ€˜π‘Š))
3938adantr 482 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})) βŠ† (Baseβ€˜π‘Š))
40 imaco 6251 . . . . . . . . . 10 ((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) = (𝐹 β€œ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})))
4112difeq1d 4122 . . . . . . . . . . . . . . 15 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}) = (𝐾 βˆ– {π‘₯}))
4241imaeq2d 6060 . . . . . . . . . . . . . 14 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) = (𝐺 β€œ (𝐾 βˆ– {π‘₯})))
43 df-f1 6549 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾–1-1β†’dom 𝐹 ↔ (𝐺:𝐾⟢dom 𝐹 ∧ Fun ◑𝐺))
4443simprbi 498 . . . . . . . . . . . . . . . 16 (𝐺:𝐾–1-1β†’dom 𝐹 β†’ Fun ◑𝐺)
45443ad2ant3 1136 . . . . . . . . . . . . . . 15 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ Fun ◑𝐺)
46 imadif 6633 . . . . . . . . . . . . . . 15 (Fun ◑𝐺 β†’ (𝐺 β€œ (𝐾 βˆ– {π‘₯})) = ((𝐺 β€œ 𝐾) βˆ– (𝐺 β€œ {π‘₯})))
4745, 46syl 17 . . . . . . . . . . . . . 14 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐺 β€œ (𝐾 βˆ– {π‘₯})) = ((𝐺 β€œ 𝐾) βˆ– (𝐺 β€œ {π‘₯})))
4842, 47eqtrd 2773 . . . . . . . . . . . . 13 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) = ((𝐺 β€œ 𝐾) βˆ– (𝐺 β€œ {π‘₯})))
4948adantr 482 . . . . . . . . . . . 12 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) = ((𝐺 β€œ 𝐾) βˆ– (𝐺 β€œ {π‘₯})))
50 fnsnfv 6971 . . . . . . . . . . . . . . 15 ((𝐺 Fn 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ {(πΊβ€˜π‘₯)} = (𝐺 β€œ {π‘₯}))
5129, 50sylan 581 . . . . . . . . . . . . . 14 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ {(πΊβ€˜π‘₯)} = (𝐺 β€œ {π‘₯}))
5251difeq2d 4123 . . . . . . . . . . . . 13 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ((𝐺 β€œ 𝐾) βˆ– {(πΊβ€˜π‘₯)}) = ((𝐺 β€œ 𝐾) βˆ– (𝐺 β€œ {π‘₯})))
53 imassrn 6071 . . . . . . . . . . . . . . 15 (𝐺 β€œ 𝐾) βŠ† ran 𝐺
546adantr 482 . . . . . . . . . . . . . . . 16 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ 𝐺:𝐾⟢dom 𝐹)
5554frnd 6726 . . . . . . . . . . . . . . 15 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ran 𝐺 βŠ† dom 𝐹)
5653, 55sstrid 3994 . . . . . . . . . . . . . 14 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ (𝐺 β€œ 𝐾) βŠ† dom 𝐹)
5756ssdifd 4141 . . . . . . . . . . . . 13 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ((𝐺 β€œ 𝐾) βˆ– {(πΊβ€˜π‘₯)}) βŠ† (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))
5852, 57eqsstrrd 4022 . . . . . . . . . . . 12 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ((𝐺 β€œ 𝐾) βˆ– (𝐺 β€œ {π‘₯})) βŠ† (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))
5949, 58eqsstrd 4021 . . . . . . . . . . 11 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) βŠ† (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))
60 imass2 6102 . . . . . . . . . . 11 ((𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) βŠ† (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}) β†’ (𝐹 β€œ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})))
6159, 60syl 17 . . . . . . . . . 10 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ (𝐹 β€œ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})))
6240, 61eqsstrid 4031 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})))
631, 22lspss 20595 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})) βŠ† (Baseβ€˜π‘Š) ∧ ((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))) β†’ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) βŠ† ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))))
6435, 39, 62, 63syl3anc 1372 . . . . . . . 8 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) βŠ† ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))))
6514, 64syldan 592 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) βŠ† ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)}))))
6665sseld 3982 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})))))
6734, 66sylbid 239 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})))))
6867adantrr 716 . . . 4 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))) β†’ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜(πΊβ€˜π‘₯))) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})))))
6927, 68mtod 197 . . 3 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))))
7069ralrimivva 3201 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ βˆ€π‘₯ ∈ dom (𝐹 ∘ 𝐺)βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))))
71 simp1 1137 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ π‘Š ∈ LMod)
72 rellindf 21363 . . . . . 6 Rel LIndF
7372brrelex1i 5733 . . . . 5 (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V)
74733ad2ant2 1135 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐹 ∈ V)
75 simp3 1139 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺:𝐾–1-1β†’dom 𝐹)
7674dmexd 7896 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ dom 𝐹 ∈ V)
77 f1dmex 7943 . . . . . 6 ((𝐺:𝐾–1-1β†’dom 𝐹 ∧ dom 𝐹 ∈ V) β†’ 𝐾 ∈ V)
7875, 76, 77syl2anc 585 . . . . 5 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐾 ∈ V)
796, 78fexd 7229 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺 ∈ V)
80 coexg 7920 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) β†’ (𝐹 ∘ 𝐺) ∈ V)
8174, 79, 80syl2anc 585 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺) ∈ V)
821, 21, 22, 23, 25, 24islindf 21367 . . 3 ((π‘Š ∈ LMod ∧ (𝐹 ∘ 𝐺) ∈ V) β†’ ((𝐹 ∘ 𝐺) LIndF π‘Š ↔ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟢(Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ dom (𝐹 ∘ 𝐺)βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))))))
8371, 81, 82syl2anc 585 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ ((𝐹 ∘ 𝐺) LIndF π‘Š ↔ ((𝐹 ∘ 𝐺):dom (𝐹 ∘ 𝐺)⟢(Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ dom (𝐹 ∘ 𝐺)βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)((𝐹 ∘ 𝐺)β€˜π‘₯)) ∈ ((LSpanβ€˜π‘Š)β€˜((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}))))))
849, 70, 83mpbir2and 712 1 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺) LIndF π‘Š)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  Vcvv 3475   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629   class class class wbr 5149  β—‘ccnv 5676  dom cdm 5677  ran crn 5678   β€œ cima 5680   ∘ ccom 5681  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€“1-1β†’wf1 6541  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  LModclmod 20471  LSpanclspn 20582   LIndF clindf 21359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-slot 17115  df-ndx 17127  df-base 17145  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-lmod 20473  df-lss 20543  df-lsp 20583  df-lindf 21361
This theorem is referenced by:  lindfres  21378  f1linds  21380
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