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Theorem linindsi 46764
Description: The implications of being a linearly independent subset. (Contributed by AV, 13-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
linindsi (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 )))
Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,π‘₯   𝑆,𝑓,π‘₯
Allowed substitution hints:   𝐡(π‘₯,𝑓)   𝑅(π‘₯,𝑓)   𝐸(π‘₯)   0 (π‘₯,𝑓)   𝑍(π‘₯,𝑓)

Proof of Theorem linindsi
StepHypRef Expression
1 linindsv 46762 . . 3 (𝑆 linIndS 𝑀 β†’ (𝑆 ∈ V ∧ 𝑀 ∈ V))
2 islininds.b . . . 4 𝐡 = (Baseβ€˜π‘€)
3 islininds.z . . . 4 𝑍 = (0gβ€˜π‘€)
4 islininds.r . . . 4 𝑅 = (Scalarβ€˜π‘€)
5 islininds.e . . . 4 𝐸 = (Baseβ€˜π‘…)
6 islininds.0 . . . 4 0 = (0gβ€˜π‘…)
72, 3, 4, 5, 6islininds 46763 . . 3 ((𝑆 ∈ V ∧ 𝑀 ∈ V) β†’ (𝑆 linIndS 𝑀 ↔ (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 ))))
81, 7syl 17 . 2 (𝑆 linIndS 𝑀 β†’ (𝑆 linIndS 𝑀 ↔ (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 ))))
98ibi 266 1 (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 )))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3060  Vcvv 3472  π’« cpw 4595   class class class wbr 5140  β€˜cfv 6531  (class class class)co 7392   ↑m cmap 8802   finSupp cfsupp 9343  Basecbs 17125  Scalarcsca 17181  0gc0g 17366   linC clinc 46721   linIndS clininds 46757
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-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5141  df-opab 5203  df-xp 5674  df-rel 5675  df-iota 6483  df-fv 6539  df-ov 7395  df-lininds 46759
This theorem is referenced by:  linindslinci  46765  linindscl  46768
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