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Theorem linindslinci 42031
 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 42030 . . 3 (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
7 breq1 4586 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓 finSupp 0𝐹 finSupp 0 ))
8 oveq1 6556 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓( linC ‘𝑀)𝑆) = (𝐹( linC ‘𝑀)𝑆))
98eqeq1d 2612 . . . . . . . . . 10 (𝑓 = 𝐹 → ((𝑓( linC ‘𝑀)𝑆) = 𝑍 ↔ (𝐹( linC ‘𝑀)𝑆) = 𝑍))
107, 9anbi12d 743 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) ↔ (𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)))
11 fveq1 6102 . . . . . . . . . . 11 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
1211eqeq1d 2612 . . . . . . . . . 10 (𝑓 = 𝐹 → ((𝑓𝑥) = 0 ↔ (𝐹𝑥) = 0 ))
1312ralbidv 2969 . . . . . . . . 9 (𝑓 = 𝐹 → (∀𝑥𝑆 (𝑓𝑥) = 0 ↔ ∀𝑥𝑆 (𝐹𝑥) = 0 ))
1410, 13imbi12d 333 . . . . . . . 8 (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 3278 . . . . . . 7 (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 84 . . . . . 6 (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 1254 . . . . 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 ))
1817com12 32 . . . 4 (∀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 ))
1918adantl 481 . . 3 ((𝑆 ∈ 𝒫 𝐵 ∧ ∀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 ))
206, 19syl 17 . 2 (𝑆 linIndS 𝑀 → ((𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝐹𝑥) = 0 ))
2120imp 444 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  𝒫 cpw 4108   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  Scalarcsca 15771  0gc0g 15923   linC clinc 41987   linIndS clininds 42023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-iota 5768  df-fv 5812  df-ov 6552  df-lininds 42025 This theorem is referenced by: (None)
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