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Theorem lindfrn 21376
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 2733 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
21lindff 21370 . . . 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 6726 . 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 6071 . . . . . . . . 9 (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† ran 𝐹
76, 4sstrid 3994 . . . . . . . 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 6722 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ Fun 𝐹)
10 eldifsn 4791 . . . . . . . . . 10 (π‘₯ ∈ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) ↔ (π‘₯ ∈ ran 𝐹 ∧ π‘₯ β‰  (πΉβ€˜π‘¦)))
11 funfn 6579 . . . . . . . . . . . . . 14 (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
12 fvelrnb 6953 . . . . . . . . . . . . . 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 4132 . . . . . . . . . . . . . . . . . 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 6892 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑦 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘¦))
2019necon3i 2974 . . . . . . . . . . . . . . . . . . 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 4791 . . . . . . . . . . . . . . . . . 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 7233 . . . . . . . . . . . . . . . 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 3004 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘˜) = π‘₯ β†’ ((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) ↔ π‘₯ β‰  (πΉβ€˜π‘¦)))
29 eleq1 2822 . . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . 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 3989 . . . . . . . 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 2733 . . . . . . . 8 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
391, 38lspss 20595 . . . . . . 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 4127 . . . . . . 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 4794 . . . . . . 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 2733 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
49 eqid 2733 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
50 eqid 2733 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
51 eqid 2733 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
5248, 38, 49, 50, 51lindfind 21371 . . . . . 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 3986 . . . 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 3201 . . 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 6580 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹 Fn dom 𝐹)
57 oveq2 7417 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘¦) β†’ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) = (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)))
58 sneq 4639 . . . . . . . . . 10 (π‘₯ = (πΉβ€˜π‘¦) β†’ {π‘₯} = {(πΉβ€˜π‘¦)})
5958difeq2d 4123 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘¦) β†’ (ran 𝐹 βˆ– {π‘₯}) = (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))
6059fveq2d 6896 . . . . . . . 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 2828 . . . . . . 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 3178 . . . . 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 7090 . . . 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 21368 . . 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 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   βŠ† wss 3949  {csn 4629   class class class wbr 5149  dom cdm 5677  ran crn 5678   β€œ cima 5680  Fun wfun 6538   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  LModclmod 20471  LSpanclspn 20582   LIndF clindf 21359  LIndSclinds 21360
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  df-linds 21362
This theorem is referenced by:  islindf3  21381  lindsmm  21383  matunitlindflem2  36485
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