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Theorem f1lindf 21596
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 2730 . . . . . . 7 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
21lindff 21589 . . . . . 6 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ LMod) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
32ancoms 457 . . . . 5 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
433adant3 1130 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
5 f1f 6786 . . . . 5 (𝐺:𝐾–1-1β†’dom 𝐹 β†’ 𝐺:𝐾⟢dom 𝐹)
653ad2ant3 1133 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺:𝐾⟢dom 𝐹)
7 fco 6740 . . . 4 ((𝐹:dom 𝐹⟢(Baseβ€˜π‘Š) ∧ 𝐺:𝐾⟢dom 𝐹) β†’ (𝐹 ∘ 𝐺):𝐾⟢(Baseβ€˜π‘Š))
84, 6, 7syl2anc 582 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺):𝐾⟢(Baseβ€˜π‘Š))
98ffdmd 6747 . 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 1190 . . . . 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 479 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ 𝐺:𝐾⟢dom 𝐹)
128fdmd 6727 . . . . . . . . 9 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ dom (𝐹 ∘ 𝐺) = 𝐾)
1312eleq2d 2817 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (π‘₯ ∈ dom (𝐹 ∘ 𝐺) ↔ π‘₯ ∈ 𝐾))
1413biimpa 475 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ π‘₯ ∈ 𝐾)
1511, 14ffvelcdmd 7086 . . . . . 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 713 . . . . 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 4125 . . . . . 6 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
1817ad2antll 725 . . . . 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 4792 . . . . . 6 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
2019ad2antll 725 . . . . 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 2730 . . . . . 6 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
22 eqid 2730 . . . . . 6 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
23 eqid 2730 . . . . . 6 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
24 eqid 2730 . . . . . 6 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
25 eqid 2730 . . . . . 6 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
2621, 22, 23, 24, 25lindfind 21590 . . . . 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 835 . . . 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 6787 . . . . . . . . . . 11 (𝐺:𝐾–1-1β†’dom 𝐹 β†’ 𝐺 Fn 𝐾)
29283ad2ant3 1133 . . . . . . . . . 10 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺 Fn 𝐾)
3029adantr 479 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ dom (𝐹 ∘ 𝐺)) β†’ 𝐺 Fn 𝐾)
31 fvco2 6987 . . . . . . . . 9 ((𝐺 Fn 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ ((𝐹 ∘ 𝐺)β€˜π‘₯) = (πΉβ€˜(πΊβ€˜π‘₯)))
3230, 14, 31syl2anc 582 . . . . . . . 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 7427 . . . . . . 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 2816 . . . . . 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 1189 . . . . . . . . 9 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ π‘Š ∈ LMod)
36 imassrn 6069 . . . . . . . . . . 11 (𝐹 β€œ (dom 𝐹 βˆ– {(πΊβ€˜π‘₯)})) βŠ† ran 𝐹
374frnd 6724 . . . . . . . . . . 11 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘Š))
3836, 37sstrid 3992 . . . . . . . . . 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 479 . . . . . . . . 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 6249 . . . . . . . . . 10 ((𝐹 ∘ 𝐺) β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})) = (𝐹 β€œ (𝐺 β€œ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯})))
4112difeq1d 4120 . . . . . . . . . . . . . . 15 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (dom (𝐹 ∘ 𝐺) βˆ– {π‘₯}) = (𝐾 βˆ– {π‘₯}))
4241imaeq2d 6058 . . . . . . . . . . . . . 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 6547 . . . . . . . . . . . . . . . . 17 (𝐺:𝐾–1-1β†’dom 𝐹 ↔ (𝐺:𝐾⟢dom 𝐹 ∧ Fun ◑𝐺))
4443simprbi 495 . . . . . . . . . . . . . . . 16 (𝐺:𝐾–1-1β†’dom 𝐹 β†’ Fun ◑𝐺)
45443ad2ant3 1133 . . . . . . . . . . . . . . 15 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ Fun ◑𝐺)
46 imadif 6631 . . . . . . . . . . . . . . 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 2770 . . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 6969 . . . . . . . . . . . . . . 15 ((𝐺 Fn 𝐾 ∧ π‘₯ ∈ 𝐾) β†’ {(πΊβ€˜π‘₯)} = (𝐺 β€œ {π‘₯}))
5129, 50sylan 578 . . . . . . . . . . . . . 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 4121 . . . . . . . . . . . . 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 6069 . . . . . . . . . . . . . . 15 (𝐺 β€œ 𝐾) βŠ† ran 𝐺
546adantr 479 . . . . . . . . . . . . . . . 16 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ 𝐺:𝐾⟢dom 𝐹)
5554frnd 6724 . . . . . . . . . . . . . . 15 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) ∧ π‘₯ ∈ 𝐾) β†’ ran 𝐺 βŠ† dom 𝐹)
5653, 55sstrid 3992 . . . . . . . . . . . . . 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 4139 . . . . . . . . . . . . 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 4020 . . . . . . . . . . . 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 4019 . . . . . . . . . . 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 6100 . . . . . . . . . . 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 4029 . . . . . . . . 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 20739 . . . . . . . . 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 1369 . . . . . . . 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 589 . . . . . . 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 3980 . . . . . 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 713 . . . 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 3198 . 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 1134 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ π‘Š ∈ LMod)
72 rellindf 21582 . . . . . 6 Rel LIndF
7372brrelex1i 5731 . . . . 5 (𝐹 LIndF π‘Š β†’ 𝐹 ∈ V)
74733ad2ant2 1132 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐹 ∈ V)
75 simp3 1136 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺:𝐾–1-1β†’dom 𝐹)
7674dmexd 7898 . . . . . 6 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ dom 𝐹 ∈ V)
77 f1dmex 7945 . . . . . 6 ((𝐺:𝐾–1-1β†’dom 𝐹 ∧ dom 𝐹 ∈ V) β†’ 𝐾 ∈ V)
7875, 76, 77syl2anc 582 . . . . 5 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐾 ∈ V)
796, 78fexd 7230 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ 𝐺 ∈ V)
80 coexg 7922 . . . 4 ((𝐹 ∈ V ∧ 𝐺 ∈ V) β†’ (𝐹 ∘ 𝐺) ∈ V)
8174, 79, 80syl2anc 582 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š ∧ 𝐺:𝐾–1-1β†’dom 𝐹) β†’ (𝐹 ∘ 𝐺) ∈ V)
821, 21, 22, 23, 25, 24islindf 21586 . . 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 582 . 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 709 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ€wral 3059  Vcvv 3472   βˆ– cdif 3944   βŠ† wss 3947  {csn 4627   class class class wbr 5147  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β€œ cima 5678   ∘ ccom 5679  Fun wfun 6536   Fn wfn 6537  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€˜cfv 6542  (class class class)co 7411  Basecbs 17148  Scalarcsca 17204   ·𝑠 cvsca 17205  0gc0g 17389  LModclmod 20614  LSpanclspn 20726   LIndF clindf 21578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12217  df-slot 17119  df-ndx 17131  df-base 17149  df-0g 17391  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-grp 18858  df-lmod 20616  df-lss 20687  df-lsp 20727  df-lindf 21580
This theorem is referenced by:  lindfres  21597  f1linds  21599
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