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Theorem islininds 41497
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 473 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
2 fveq2 6150 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 islininds.b . . . . . . 7 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2678 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
54adantl 482 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘𝑚) = 𝐵)
65pweqd 4140 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
71, 6eleq12d 2698 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵))
8 fveq2 6150 . . . . . . . . 9 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
9 islininds.r . . . . . . . . 9 𝑅 = (Scalar‘𝑀)
108, 9syl6eqr 2678 . . . . . . . 8 (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅)
1110fveq2d 6154 . . . . . . 7 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅))
12 islininds.e . . . . . . 7 𝐸 = (Base‘𝑅)
1311, 12syl6eqr 2678 . . . . . 6 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸)
1413adantl 482 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸)
1514, 1oveq12d 6623 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑠) = (𝐸𝑚 𝑆))
168adantl 482 . . . . . . . . . 10 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀))
1716, 9syl6eqr 2678 . . . . . . . . 9 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅)
1817fveq2d 6154 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = (0g𝑅))
19 islininds.0 . . . . . . . 8 0 = (0g𝑅)
2018, 19syl6eqr 2678 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
2120breq2d 4630 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 ))
22 fveq2 6150 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
2322adantl 482 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀))
24 eqidd 2627 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑓 = 𝑓)
2523, 24, 1oveq123d 6626 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆))
26 fveq2 6150 . . . . . . . . 9 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
2726adantl 482 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = (0g𝑀))
28 islininds.z . . . . . . . 8 𝑍 = (0g𝑀)
2927, 28syl6eqr 2678 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = 𝑍)
3025, 29eqeq12d 2641 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓( linC ‘𝑚)𝑠) = (0g𝑚) ↔ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
3121, 30anbi12d 746 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
3210fveq2d 6154 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑅))
3332, 19syl6eqr 2678 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = 0 )
3433adantl 482 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
3534eqeq2d 2636 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ (𝑓𝑥) = 0 ))
361, 35raleqbidv 3146 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥𝑆 (𝑓𝑥) = 0 ))
3731, 36imbi12d 334 . . . 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 3146 . . 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 746 . 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 41493 . 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 4954 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 196  wa 384   = wceq 1480  wcel 1992  wral 2912  𝒫 cpw 4135   class class class wbr 4618  cfv 5850  (class class class)co 6605  𝑚 cmap 7803   finSupp cfsupp 8220  Basecbs 15776  Scalarcsca 15860  0gc0g 16016   linC clinc 41455   linIndS clininds 41491
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-iota 5813  df-fv 5858  df-ov 6608  df-lininds 41493
This theorem is referenced by:  linindsi  41498  islinindfis  41500  islindeps  41504  lindslininds  41515  linds0  41516  lindsrng01  41519  snlindsntor  41522
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