Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islininds Structured version   Visualization version   GIF version

Theorem islininds 42763
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 468 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑠 = 𝑆)
2 fveq2 6332 . . . . . . 7 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3 islininds.b . . . . . . 7 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2823 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = 𝐵)
54adantl 467 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘𝑚) = 𝐵)
65pweqd 4302 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝒫 (Base‘𝑚) = 𝒫 𝐵)
71, 6eleq12d 2844 . . 3 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑠 ∈ 𝒫 (Base‘𝑚) ↔ 𝑆 ∈ 𝒫 𝐵))
8 fveq2 6332 . . . . . . . . 9 (𝑚 = 𝑀 → (Scalar‘𝑚) = (Scalar‘𝑀))
9 islininds.r . . . . . . . . 9 𝑅 = (Scalar‘𝑀)
108, 9syl6eqr 2823 . . . . . . . 8 (𝑚 = 𝑀 → (Scalar‘𝑚) = 𝑅)
1110fveq2d 6336 . . . . . . 7 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = (Base‘𝑅))
12 islininds.e . . . . . . 7 𝐸 = (Base‘𝑅)
1311, 12syl6eqr 2823 . . . . . 6 (𝑚 = 𝑀 → (Base‘(Scalar‘𝑚)) = 𝐸)
1413adantl 467 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (Base‘(Scalar‘𝑚)) = 𝐸)
1514, 1oveq12d 6811 . . . 4 ((𝑠 = 𝑆𝑚 = 𝑀) → ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑠) = (𝐸𝑚 𝑆))
168adantl 467 . . . . . . . . . 10 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = (Scalar‘𝑀))
1716, 9syl6eqr 2823 . . . . . . . . 9 ((𝑠 = 𝑆𝑚 = 𝑀) → (Scalar‘𝑚) = 𝑅)
1817fveq2d 6336 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = (0g𝑅))
19 islininds.0 . . . . . . . 8 0 = (0g𝑅)
2018, 19syl6eqr 2823 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
2120breq2d 4798 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓 finSupp (0g‘(Scalar‘𝑚)) ↔ 𝑓 finSupp 0 ))
22 fveq2 6332 . . . . . . . . 9 (𝑚 = 𝑀 → ( linC ‘𝑚) = ( linC ‘𝑀))
2322adantl 467 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → ( linC ‘𝑚) = ( linC ‘𝑀))
24 eqidd 2772 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → 𝑓 = 𝑓)
2523, 24, 1oveq123d 6814 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (𝑓( linC ‘𝑚)𝑠) = (𝑓( linC ‘𝑀)𝑆))
26 fveq2 6332 . . . . . . . . 9 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
2726adantl 467 . . . . . . . 8 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = (0g𝑀))
28 islininds.z . . . . . . . 8 𝑍 = (0g𝑀)
2927, 28syl6eqr 2823 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g𝑚) = 𝑍)
3025, 29eqeq12d 2786 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓( linC ‘𝑚)𝑠) = (0g𝑚) ↔ (𝑓( linC ‘𝑀)𝑆) = 𝑍))
3121, 30anbi12d 616 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) ↔ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)))
3210fveq2d 6336 . . . . . . . . 9 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = (0g𝑅))
3332, 19syl6eqr 2823 . . . . . . . 8 (𝑚 = 𝑀 → (0g‘(Scalar‘𝑚)) = 0 )
3433adantl 467 . . . . . . 7 ((𝑠 = 𝑆𝑚 = 𝑀) → (0g‘(Scalar‘𝑚)) = 0 )
3534eqeq2d 2781 . . . . . 6 ((𝑠 = 𝑆𝑚 = 𝑀) → ((𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ (𝑓𝑥) = 0 ))
361, 35raleqbidv 3301 . . . . 5 ((𝑠 = 𝑆𝑚 = 𝑀) → (∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚)) ↔ ∀𝑥𝑆 (𝑓𝑥) = 0 ))
3731, 36imbi12d 333 . . . 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 3301 . . 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 616 . 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 42759 . 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 5122 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 382   = wceq 1631  wcel 2145  wral 3061  𝒫 cpw 4297   class class class wbr 4786  cfv 6031  (class class class)co 6793  𝑚 cmap 8009   finSupp cfsupp 8431  Basecbs 16064  Scalarcsca 16152  0gc0g 16308   linC clinc 42721   linIndS clininds 42757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-iota 5994  df-fv 6039  df-ov 6796  df-lininds 42759
This theorem is referenced by:  linindsi  42764  islinindfis  42766  islindeps  42770  lindslininds  42781  linds0  42782  lindsrng01  42785  snlindsntor  42788
  Copyright terms: Public domain W3C validator