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Theorem linindslinci 47129
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 𝐡 = (Baseβ€˜π‘€)
islininds.z 𝑍 = (0gβ€˜π‘€)
islininds.r 𝑅 = (Scalarβ€˜π‘€)
islininds.e 𝐸 = (Baseβ€˜π‘…)
islininds.0 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
linindslinci ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍)) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 )
Distinct variable groups:   π‘₯,𝑀   π‘₯,𝑆   π‘₯,𝐹
Allowed substitution hints:   𝐡(π‘₯)   𝑅(π‘₯)   𝐸(π‘₯)   0 (π‘₯)   𝑍(π‘₯)

Proof of Theorem linindslinci
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 islininds.b . . . 4 𝐡 = (Baseβ€˜π‘€)
2 islininds.z . . . 4 𝑍 = (0gβ€˜π‘€)
3 islininds.r . . . 4 𝑅 = (Scalarβ€˜π‘€)
4 islininds.e . . . 4 𝐸 = (Baseβ€˜π‘…)
5 islininds.0 . . . 4 0 = (0gβ€˜π‘…)
61, 2, 3, 4, 5linindsi 47128 . . 3 (𝑆 linIndS 𝑀 β†’ (𝑆 ∈ 𝒫 𝐡 ∧ βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 )))
7 breq1 5152 . . . . . . . . 9 (𝑓 = 𝐹 β†’ (𝑓 finSupp 0 ↔ 𝐹 finSupp 0 ))
8 oveq1 7416 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ (𝑓( linC β€˜π‘€)𝑆) = (𝐹( linC β€˜π‘€)𝑆))
98eqeq1d 2735 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((𝑓( linC β€˜π‘€)𝑆) = 𝑍 ↔ (𝐹( linC β€˜π‘€)𝑆) = 𝑍))
107, 9anbi12d 632 . . . . . . . 8 (𝑓 = 𝐹 β†’ ((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) ↔ (𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍)))
11 fveq1 6891 . . . . . . . . . 10 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
1211eqeq1d 2735 . . . . . . . . 9 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯) = 0 ↔ (πΉβ€˜π‘₯) = 0 ))
1312ralbidv 3178 . . . . . . . 8 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ↔ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 ))
1410, 13imbi12d 345 . . . . . . 7 (𝑓 = 𝐹 β†’ (((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) ↔ ((𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 )))
1514rspcv 3609 . . . . . 6 (𝐹 ∈ (𝐸 ↑m 𝑆) β†’ (βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ ((𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 )))
1615com23 86 . . . . 5 (𝐹 ∈ (𝐸 ↑m 𝑆) β†’ ((𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍) β†’ (βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 )))
17163impib 1117 . . . 4 ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍) β†’ (βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 ))
1817com12 32 . . 3 (βˆ€π‘“ ∈ (𝐸 ↑m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (π‘“β€˜π‘₯) = 0 ) β†’ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 ))
196, 18simpl2im 505 . 2 (𝑆 linIndS 𝑀 β†’ ((𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 ))
2019imp 408 1 ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸 ↑m 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC β€˜π‘€)𝑆) = 𝑍)) β†’ βˆ€π‘₯ ∈ 𝑆 (πΉβ€˜π‘₯) = 0 )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  π’« cpw 4603   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409   ↑m cmap 8820   finSupp cfsupp 9361  Basecbs 17144  Scalarcsca 17200  0gc0g 17385   linC clinc 47085   linIndS clininds 47121
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-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-iota 6496  df-fv 6552  df-ov 7412  df-lininds 47123
This theorem is referenced by: (None)
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