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Theorem islininds 48292
Description: The property of being a linearly independent subset. (Contributed by AV, 13-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
islininds ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
Distinct variable groups:   𝑓,𝐸   𝑓,𝑀,𝑥   𝑆,𝑓,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑓)   𝑅(𝑥,𝑓)   𝐸(𝑥)   𝑉(𝑥,𝑓)   𝑊(𝑥,𝑓)   0 (𝑥,𝑓)   𝑍(𝑥,𝑓)

Proof of Theorem islininds
Dummy variables 𝑚 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 482 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
2 fveq2 6907 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 islininds.b . . . . . . 7 𝐵 = (Base‘𝑀)
42, 3eqtr4di 2793 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
54adantl 481 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘𝑚) = 𝐵)
65pweqd 4622 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
71, 6eleq12d 2833 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵))
8 fveq2 6907 . . . . . . . . 9 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
9 islininds.r . . . . . . . . 9 𝑅 = (Scalar‘𝑀)
108, 9eqtr4di 2793 . . . . . . . 8 (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅)
1110fveq2d 6911 . . . . . . 7 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅))
12 islininds.e . . . . . . 7 𝐸 = (Base‘𝑅)
1311, 12eqtr4di 2793 . . . . . 6 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸)
1413adantl 481 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸)
1514, 1oveq12d 7449 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑m 𝑠) = (𝐸m 𝑆))
168adantl 481 . . . . . . . . . 10 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀))
1716, 9eqtr4di 2793 . . . . . . . . 9 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅)
1817fveq2d 6911 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = (0g𝑅))
19 islininds.0 . . . . . . . 8 0 = (0g𝑅)
2018, 19eqtr4di 2793 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
2120breq2d 5160 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 ))
22 fveq2 6907 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
2322adantl 481 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀))
24 eqidd 2736 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑓 = 𝑓)
2523, 24, 1oveq123d 7452 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆))
26 fveq2 6907 . . . . . . . . 9 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
2726adantl 481 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = (0g𝑀))
28 islininds.z . . . . . . . 8 𝑍 = (0g𝑀)
2927, 28eqtr4di 2793 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = 𝑍)
3025, 29eqeq12d 2751 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓( linC ‘𝑚)𝑠) = (0g𝑚) ↔ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
3121, 30anbi12d 632 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
3210fveq2d 6911 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑅))
3332, 19eqtr4di 2793 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = 0 )
3433adantl 481 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
3534eqeq2d 2746 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ (𝑓𝑥) = 0 ))
361, 35raleqbidv 3344 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥𝑆 (𝑓𝑥) = 0 ))
3731, 36imbi12d 344 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → (((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))) ↔ ((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
3815, 37raleqbidv 3344 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))) ↔ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))
397, 38anbi12d 632 . 2 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)))) ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
40 df-lininds 48288 . 2 linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))))}
4139, 40brabga 5544 1 ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸m 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  𝒫 cpw 4605   class class class wbr 5148  cfv 6563  (class class class)co 7431  m cmap 8865   finSupp cfsupp 9399  Basecbs 17245  Scalarcsca 17301  0gc0g 17486   linC clinc 48250   linIndS clininds 48286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571  df-ov 7434  df-lininds 48288
This theorem is referenced by:  linindsi  48293  islinindfis  48295  islindeps  48299  lindslininds  48310  linds0  48311  lindsrng01  48314  snlindsntor  48317
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