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Theorem lindfrn 21367
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 2732 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
21lindff 21361 . . . 4 ((𝐹 LIndF π‘Š ∧ π‘Š ∈ LMod) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
32ancoms 459 . . 3 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹:dom 𝐹⟢(Baseβ€˜π‘Š))
43frnd 6722 . 2 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ ran 𝐹 βŠ† (Baseβ€˜π‘Š))
5 simpll 765 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ 𝑦 ∈ dom 𝐹) β†’ π‘Š ∈ LMod)
6 imassrn 6068 . . . . . . . . 9 (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† ran 𝐹
76, 4sstrid 3992 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† (Baseβ€˜π‘Š))
87adantr 481 . . . . . . 7 (((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) ∧ 𝑦 ∈ dom 𝐹) β†’ (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})) βŠ† (Baseβ€˜π‘Š))
93ffund 6718 . . . . . . . 8 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ Fun 𝐹)
10 eldifsn 4789 . . . . . . . . . 10 (π‘₯ ∈ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) ↔ (π‘₯ ∈ ran 𝐹 ∧ π‘₯ β‰  (πΉβ€˜π‘¦)))
11 funfn 6575 . . . . . . . . . . . . . 14 (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
12 fvelrnb 6949 . . . . . . . . . . . . . 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 481 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (π‘₯ ∈ ran 𝐹 ↔ βˆƒπ‘˜ ∈ dom 𝐹(πΉβ€˜π‘˜) = π‘₯))
15 difss 4130 . . . . . . . . . . . . . . . . . 18 (dom 𝐹 βˆ– {𝑦}) βŠ† dom 𝐹
1615jctr 525 . . . . . . . . . . . . . . . . 17 (Fun 𝐹 β†’ (Fun 𝐹 ∧ (dom 𝐹 βˆ– {𝑦}) βŠ† dom 𝐹))
1716ad2antrr 724 . . . . . . . . . . . . . . . 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 483 . . . . . . . . . . . . . . . . . 18 ((π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦)) β†’ π‘˜ ∈ dom 𝐹)
19 fveq2 6888 . . . . . . . . . . . . . . . . . . . 20 (π‘˜ = 𝑦 β†’ (πΉβ€˜π‘˜) = (πΉβ€˜π‘¦))
2019necon3i 2973 . . . . . . . . . . . . . . . . . . 19 ((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) β†’ π‘˜ β‰  𝑦)
2120adantl 482 . . . . . . . . . . . . . . . . . 18 ((π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦)) β†’ π‘˜ β‰  𝑦)
22 eldifsn 4789 . . . . . . . . . . . . . . . . . 18 (π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}) ↔ (π‘˜ ∈ dom 𝐹 ∧ π‘˜ β‰  𝑦))
2318, 21, 22sylanbrc 583 . . . . . . . . . . . . . . . . 17 ((π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦)) β†’ π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}))
2423adantl 482 . . . . . . . . . . . . . . . 16 (((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) ∧ (π‘˜ ∈ dom 𝐹 ∧ (πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦))) β†’ π‘˜ ∈ (dom 𝐹 βˆ– {𝑦}))
25 funfvima2 7229 . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . 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 3003 . . . . . . . . . . . . . . 15 ((πΉβ€˜π‘˜) = π‘₯ β†’ ((πΉβ€˜π‘˜) β‰  (πΉβ€˜π‘¦) ↔ π‘₯ β‰  (πΉβ€˜π‘¦)))
29 eleq1 2821 . . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . 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 411 . . . . . . . . . 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 3987 . . . . . . . 8 ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) β†’ (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}) βŠ† (𝐹 β€œ (dom 𝐹 βˆ– {𝑦})))
379, 36sylan 580 . . . . . . 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 2732 . . . . . . . 8 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
391, 38lspss 20587 . . . . . . 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 1371 . . . . . 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 715 . . . . 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 767 . . . . . 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 769 . . . . . 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 4125 . . . . . . 7 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ ∈ (Baseβ€˜(Scalarβ€˜π‘Š)))
4544ad2antll 727 . . . . . 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 4792 . . . . . . 7 (π‘˜ ∈ ((Baseβ€˜(Scalarβ€˜π‘Š)) βˆ– {(0gβ€˜(Scalarβ€˜π‘Š))}) β†’ π‘˜ β‰  (0gβ€˜(Scalarβ€˜π‘Š)))
4746ad2antll 727 . . . . . 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 2732 . . . . . . 7 ( ·𝑠 β€˜π‘Š) = ( ·𝑠 β€˜π‘Š)
49 eqid 2732 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
50 eqid 2732 . . . . . . 7 (0gβ€˜(Scalarβ€˜π‘Š)) = (0gβ€˜(Scalarβ€˜π‘Š))
51 eqid 2732 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
5248, 38, 49, 50, 51lindfind 21362 . . . . . 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 837 . . . . 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 3984 . . . 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 3200 . . 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 6576 . . . 4 ((π‘Š ∈ LMod ∧ 𝐹 LIndF π‘Š) β†’ 𝐹 Fn dom 𝐹)
57 oveq2 7413 . . . . . . . 8 (π‘₯ = (πΉβ€˜π‘¦) β†’ (π‘˜( ·𝑠 β€˜π‘Š)π‘₯) = (π‘˜( ·𝑠 β€˜π‘Š)(πΉβ€˜π‘¦)))
58 sneq 4637 . . . . . . . . . 10 (π‘₯ = (πΉβ€˜π‘¦) β†’ {π‘₯} = {(πΉβ€˜π‘¦)})
5958difeq2d 4121 . . . . . . . . 9 (π‘₯ = (πΉβ€˜π‘¦) β†’ (ran 𝐹 βˆ– {π‘₯}) = (ran 𝐹 βˆ– {(πΉβ€˜π‘¦)}))
6059fveq2d 6892 . . . . . . . 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 2827 . . . . . . 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 317 . . . . . 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 3177 . . . . 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 7086 . . . 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 256 . 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 21359 . . 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 481 . 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 711 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 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3944   βŠ† wss 3947  {csn 4627   class class class wbr 5147  dom cdm 5675  ran crn 5676   β€œ cima 5678  Fun wfun 6534   Fn wfn 6535  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  Scalarcsca 17196   ·𝑠 cvsca 17197  0gc0g 17381  LModclmod 20463  LSpanclspn 20574   LIndF clindf 21350  LIndSclinds 21351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-1cn 11164  ax-addcl 11166
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-om 7852  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-nn 12209  df-slot 17111  df-ndx 17123  df-base 17141  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-lmod 20465  df-lss 20535  df-lsp 20575  df-lindf 21352  df-linds 21353
This theorem is referenced by:  islindf3  21372  lindsmm  21374  matunitlindflem2  36473
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