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Theorem linindslinci 44431
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 44430 . . 3 (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
7 breq1 5060 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓 finSupp 0𝐹 finSupp 0 ))
8 oveq1 7152 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓( linC ‘𝑀)𝑆) = (𝐹( linC ‘𝑀)𝑆))
98eqeq1d 2820 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ (𝐹( linC ‘𝑀)𝑆) = 𝑍))
107, 9anbi12d 630 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)))
11 fveq1 6662 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
1211eqeq1d 2820 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) = 0 ↔ (𝐹𝑥) = 0 ))
1312ralbidv 3194 . . . . . . . 8 (𝑓 = 𝐹 → (∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∀𝑥𝑆 (𝐹𝑥) = 0 ))
1410, 13imbi12d 346 . . . . . . 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 3615 . . . . . 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 1108 . . . 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 504 . 2 (𝑆 linIndS 𝑀 → ((𝐹 ∈ (𝐸m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝐹𝑥) = 0 ))
2019imp 407 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 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  𝒫 cpw 4535   class class class wbr 5057  cfv 6348  (class class class)co 7145  m cmap 8395   finSupp cfsupp 8821  Basecbs 16471  Scalarcsca 16556  0gc0g 16701   linC clinc 44387   linIndS clininds 44423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-iota 6307  df-fv 6356  df-ov 7148  df-lininds 44425
This theorem is referenced by: (None)
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