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Theorem lindfrn 21243
Description: The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Assertion
Ref Expression
lindfrn ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ ran 𝐹 ∈ (LIndSβ€˜π‘Š))

Proof of Theorem lindfrn
Dummy variables π‘˜ π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
21lindff 21237 . . . 4 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ LMod) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
32ancoms 460 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
43frnd 6681 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘Š))
5 simpll 766 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ 𝑦 ∈ dom 𝐹) β†’ π‘Š ∈ LMod)
6 imassrn 6029 . . . . . . . . 9 (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† ran 𝐹
76, 4sstrid 3960 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† (Baseβ€˜π‘Š))
87adantr 482 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ 𝑦 ∈ dom 𝐹) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† (Baseβ€˜π‘Š))
93ffund 6677 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ Fun 𝐹)
10 eldifsn 4752 . . . . . . . . . 10 (π‘₯ ∈ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) ↔ (π‘₯ ∈ ran 𝐹 ∧ π‘₯ β‰  (πΉβ€˜π‘¦)))
11 funfn 6536 . . . . . . . . . . . . . 14 (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
12 fvelrnb 6908 . . . . . . . . . . . . . 14 (𝐹 Fn dom 𝐹 β†’ (π‘₯ ∈ ran 𝐹 ↔ βˆƒπ‘˜ ∈ dom 𝐹(πΉβ€˜π‘˜) = π‘₯))
1311, 12sylbi 216 . . . . . . . . . . . . 13 (Fun 𝐹 β†’ (π‘₯ ∈ ran 𝐹 ↔ βˆƒπ‘˜ ∈ dom 𝐹(πΉβ€˜π‘˜) = π‘₯))
1413adantr 482 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (π‘₯ ∈ ran 𝐹 ↔ βˆƒπ‘˜ ∈ dom 𝐹(πΉβ€˜π‘˜) = π‘₯))
15 difss 4096 . . . . . . . . . . . . . . . . . 18 (dom 𝐹 βˆ– {𝑦}) βŠ† dom 𝐹
1615jctr 526 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 β†’ (Fun 𝐹 ∧ (dom 𝐹 βˆ– {𝑦}) βŠ† dom 𝐹))
1716ad2antrr 725 . . . . . . . . . . . . . . . 16 (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦))) β†’ (Fun 𝐹 ∧ (dom 𝐹 βˆ– {𝑦}) βŠ† dom 𝐹))
18 simpl 484 . . . . . . . . . . . . . . . . . 18 ((π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦)) β†’ π‘˜ ∈ dom 𝐹)
19 fveq2 6847 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑦 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘¦))
2019necon3i 2977 . . . . . . . . . . . . . . . . . . 19 ((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) β†’ π‘˜ β‰  𝑦)
2120adantl 483 . . . . . . . . . . . . . . . . . 18 ((π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦)) β†’ π‘˜ β‰  𝑦)
22 eldifsn 4752 . . . . . . . . . . . . . . . . . 18 (π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}) ↔ (π‘˜ ∈ dom 𝐹 ∧ π‘˜ β‰  𝑦))
2318, 21, 22sylanbrc 584 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦)) β†’ π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}))
2423adantl 483 . . . . . . . . . . . . . . . 16 (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦))) β†’ π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}))
25 funfvima2 7186 . . . . . . . . . . . . . . . 16 ((Fun 𝐹 ∧ (dom 𝐹 βˆ– {𝑦}) βŠ† dom 𝐹) β†’ (π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}) β†’ (πΉβ€˜π‘˜) ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
2617, 24, 25sylc 65 . . . . . . . . . . . . . . 15 (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦))) β†’ (πΉβ€˜π‘˜) ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))
2726expr 458 . . . . . . . . . . . . . 14 (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ π‘˜ ∈ dom 𝐹) β†’ ((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) β†’ (πΉβ€˜π‘˜) ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
28 neeq1 3007 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘˜) = π‘₯ β†’ ((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) ↔ π‘₯ β‰  (πΉβ€˜π‘¦)))
29 eleq1 2826 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘˜) = π‘₯ β†’ ((πΉβ€˜π‘˜) ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) ↔ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
3028, 29imbi12d 345 . . . . . . . . . . . . . 14 ((πΉβ€˜π‘˜) = π‘₯ β†’ (((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) β†’ (πΉβ€˜π‘˜) ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))) ↔ (π‘₯ β‰  (πΉβ€˜π‘¦) β†’ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))))
3127, 30syl5ibcom 244 . . . . . . . . . . . . 13 (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ π‘˜ ∈ dom 𝐹) β†’ ((πΉβ€˜π‘˜) = π‘₯ β†’ (π‘₯ β‰  (πΉβ€˜π‘¦) β†’ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))))
3231rexlimdva 3153 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (βˆƒπ‘˜ ∈ dom 𝐹(πΉβ€˜π‘˜) = π‘₯ β†’ (π‘₯ β‰  (πΉβ€˜π‘¦) β†’ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))))
3314, 32sylbid 239 . . . . . . . . . . 11 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (π‘₯ ∈ ran 𝐹 β†’ (π‘₯ β‰  (πΉβ€˜π‘¦) β†’ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))))
3433impd 412 . . . . . . . . . 10 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ ((π‘₯ ∈ ran 𝐹 ∧ π‘₯ β‰  (πΉβ€˜π‘¦)) β†’ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
3510, 34biimtrid 241 . . . . . . . . 9 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (π‘₯ ∈ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) β†’ π‘₯ ∈ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
3635ssrdv 3955 . . . . . . . 8 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))
379, 36sylan 581 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ 𝑦 ∈ dom 𝐹) β†’ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))
38 eqid 2737 . . . . . . . 8 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
391, 38lspss 20461 . . . . . . 7 ((π‘Š ∈ LMod ∧ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† (Baseβ€˜π‘Š) ∧ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))) β†’ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)})) βŠ† ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
405, 8, 37, 39syl3anc 1372 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ 𝑦 ∈ dom 𝐹) β†’ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)})) βŠ† ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
4140adantrr 716 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)})) βŠ† ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
42 simplr 768 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝐹 LIndF π‘Š)
43 simprl 770 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ 𝑦 ∈ dom 𝐹)
44 eldifi 4091 . . . . . . 7 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
4544ad2antll 728 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
46 eldifsni 4755 . . . . . . 7 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
4746ad2antll 728 . . . . . 6 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
48 eqid 2737 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
49 eqid 2737 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
50 eqid 2737 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
51 eqid 2737 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
5248, 38, 49, 50, 51lindfind 21238 . . . . . 6 (((𝐹 LIndF π‘Š ∧ 𝑦 ∈ dom 𝐹) ∧ (π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)) ∧ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
5342, 43, 45, 47, 52syl22anc 838 . . . . 5 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(𝐹 β€œ (dom 𝐹 βˆ– {𝑦}))))
5441, 53ssneldd 3952 . . . 4 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ (𝑦 ∈ dom 𝐹 ∧ π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}))) β†’ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)})))
5554ralrimivva 3198 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ βˆ€π‘¦ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)})))
569funfnd 6537 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹 Fn dom 𝐹)
57 oveq2 7370 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘¦) β†’ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) = (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)))
58 sneq 4601 . . . . . . . . . 10 (π‘₯ = (πΉβ€˜π‘¦) β†’ {π‘₯} = {(πΉβ€˜π‘¦)})
5958difeq2d 4087 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘¦) β†’ (ran 𝐹 βˆ– {π‘₯}) = (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))
6059fveq2d 6851 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘¦) β†’ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})) = ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)})))
6157, 60eleq12d 2832 . . . . . . 7 (π‘₯ = (πΉβ€˜π‘¦) β†’ ((π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})) ↔ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))))
6261notbid 318 . . . . . 6 (π‘₯ = (πΉβ€˜π‘¦) β†’ (Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})) ↔ Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))))
6362ralbidv 3175 . . . . 5 (π‘₯ = (πΉβ€˜π‘¦) β†’ (βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})) ↔ βˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))))
6463ralrn 7043 . . . 4 (𝐹 Fn dom 𝐹 β†’ (βˆ€π‘₯ ∈ ran πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})) ↔ βˆ€π‘¦ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))))
6556, 64syl 17 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ (βˆ€π‘₯ ∈ ran πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})) ↔ βˆ€π‘¦ ∈ dom πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))))
6655, 65mpbird 257 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ βˆ€π‘₯ ∈ ran πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})))
671, 48, 38, 49, 51, 50islinds2 21235 . . 3 (π‘Š ∈ LMod β†’ (ran 𝐹 ∈ (LIndSβ€˜π‘Š) ↔ (ran 𝐹 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ ran πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})))))
6867adantr 482 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ (ran 𝐹 ∈ (LIndSβ€˜π‘Š) ↔ (ran 𝐹 βŠ† (Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ ran πΉβˆ€π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) Β¬ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) ∈ ((LSpanβ€˜π‘Š)β€˜(ran 𝐹 βˆ– {π‘₯})))))
694, 66, 68mpbir2and 712 1 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ ran 𝐹 ∈ (LIndSβ€˜π‘Š))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074   βˆ– cdif 3912   βŠ† wss 3915  {csn 4591   class class class wbr 5110  dom cdm 5638  ran crn 5639   β€œ cima 5641  Fun wfun 6495   Fn wfn 6496  βŸΆwf 6497  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  Scalarcsca 17143   ·𝑠 cvsca 17144  0gc0g 17328  LModclmod 20338  LSpanclspn 20448   LIndF clindf 21226  LIndSclinds 21227
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-1cn 11116  ax-addcl 11118
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-nn 12161  df-slot 17061  df-ndx 17073  df-base 17091  df-0g 17330  df-mgm 18504  df-sgrp 18553  df-mnd 18564  df-grp 18758  df-lmod 20340  df-lss 20409  df-lsp 20449  df-lindf 21228  df-linds 21229
This theorem is referenced by:  islindf3  21248  lindsmm  21250  matunitlindflem2  36104
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