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Theorem linindslinci 44679
Description: The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
Hypotheses
Ref Expression
islininds.b 𝐵 = (Base‘𝑀)
islininds.z 𝑍 = (0g𝑀)
islininds.r 𝑅 = (Scalar‘𝑀)
islininds.e 𝐸 = (Base‘𝑅)
islininds.0 0 = (0g𝑅)
Assertion
Ref Expression
linindslinci ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥𝑆 (𝐹𝑥) = 0 )
Distinct variable groups:   𝑥,𝑀   𝑥,𝑆   𝑥,𝐹
Allowed substitution hints:   𝐵(𝑥)   𝑅(𝑥)   𝐸(𝑥)   0 (𝑥)   𝑍(𝑥)

Proof of Theorem linindslinci
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 islininds.b . . . 4 𝐵 = (Base‘𝑀)
2 islininds.z . . . 4 𝑍 = (0g𝑀)
3 islininds.r . . . 4 𝑅 = (Scalar‘𝑀)
4 islininds.e . . . 4 𝐸 = (Base‘𝑅)
5 islininds.0 . . . 4 0 = (0g𝑅)
61, 2, 3, 4, 5linindsi 44678 . . 3 (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
7 breq1 5042 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 finSupp 0𝐹 finSupp 0 ))
8 oveq1 7137 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓( linC ‘𝑀)𝑆) = (𝐹( linC ‘𝑀)𝑆))
98eqeq1d 2823 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ (𝐹( linC ‘𝑀)𝑆) = 𝑍))
107, 9anbi12d 633 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)))
11 fveq1 6642 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
1211eqeq1d 2823 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) = 0 ↔ (𝐹𝑥) = 0 ))
1312ralbidv 3185 . . . . . . . 8 (𝑓 = 𝐹 → (∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∀𝑥𝑆 (𝐹𝑥) = 0 ))
1410, 13imbi12d 348 . . . . . . 7 (𝑓 = 𝐹 → (((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) ↔ ((𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝐹𝑥) = 0 )))
1514rspcv 3595 . . . . . 6 (𝐹 ∈ (𝐸m 𝑆) → (∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝐹𝑥) = 0 )))
1615com23 86 . . . . 5 (𝐹 ∈ (𝐸m 𝑆) → ((𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ∀𝑥𝑆 (𝐹𝑥) = 0 )))
17163impib 1113 . . . 4 ((𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ∀𝑥𝑆 (𝐹𝑥) = 0 ))
1817com12 32 . . 3 (∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ) → ((𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝐹𝑥) = 0 ))
196, 18simpl2im 507 . 2 (𝑆 linIndS 𝑀 → ((𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝐹𝑥) = 0 ))
2019imp 410 1 ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥𝑆 (𝐹𝑥) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3126  𝒫 cpw 4512   class class class wbr 5039  cfv 6328  (class class class)co 7130  m cmap 8381   finSupp cfsupp 8809  Basecbs 16462  Scalarcsca 16547  0gc0g 16692   linC clinc 44635   linIndS clininds 44671
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-iota 6287  df-fv 6336  df-ov 7133  df-lininds 44673
This theorem is referenced by: (None)
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