Users' Mathboxes Mathbox for Steven Nguyen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prjspval Structured version   Visualization version   GIF version

Theorem prjspval 42590
Description: Value of the projective space function, which is also known as the projectivization of 𝑉. (Contributed by Steven Nguyen, 29-Apr-2023.)
Hypotheses
Ref Expression
prjspval.b 𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})
prjspval.x · = ( ·𝑠𝑉)
prjspval.s 𝑆 = (Scalar‘𝑉)
prjspval.k 𝐾 = (Base‘𝑆)
Assertion
Ref Expression
prjspval (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
Distinct variable group:   𝑥,𝑙,𝑦,𝑉
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑙)   𝑆(𝑥,𝑦,𝑙)   · (𝑥,𝑦,𝑙)   𝐾(𝑥,𝑦,𝑙)

Proof of Theorem prjspval
Dummy variables 𝑏 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6920 . . . . 5 (Base‘𝑣) ∈ V
21difexi 5336 . . . 4 ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V)
4 fveq2 6907 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 fveq2 6907 . . . . . . . . . 10 (𝑣 = 𝑉 → (0g𝑣) = (0g𝑉))
65sneqd 4643 . . . . . . . . 9 (𝑣 = 𝑉 → {(0g𝑣)} = {(0g𝑉)})
74, 6difeq12d 4137 . . . . . . . 8 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = ((Base‘𝑉) ∖ {(0g𝑉)}))
8 prjspval.b . . . . . . . 8 𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})
97, 8eqtr4di 2793 . . . . . . 7 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = 𝐵)
109eqeq2d 2746 . . . . . 6 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) ↔ 𝑏 = 𝐵))
1110biimpd 229 . . . . 5 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) → 𝑏 = 𝐵))
1211imp 406 . . . 4 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → 𝑏 = 𝐵)
1311imdistani 568 . . . . . 6 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → (𝑣 = 𝑉𝑏 = 𝐵))
14 eleq2 2828 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥𝑏𝑥𝐵))
15 eleq2 2828 . . . . . . . 8 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1614, 15anbi12d 632 . . . . . . 7 (𝑏 = 𝐵 → ((𝑥𝑏𝑦𝑏) ↔ (𝑥𝐵𝑦𝐵)))
17 fveq2 6907 . . . . . . . . . . 11 (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑉)
1917, 18eqtr4di 2793 . . . . . . . . . 10 (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
2019fveq2d 6911 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21 prjspval.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
2220, 21eqtr4di 2793 . . . . . . . 8 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23 fveq2 6907 . . . . . . . . . . 11 (𝑣 = 𝑉 → ( ·𝑠𝑣) = ( ·𝑠𝑉))
24 prjspval.x . . . . . . . . . . 11 · = ( ·𝑠𝑉)
2523, 24eqtr4di 2793 . . . . . . . . . 10 (𝑣 = 𝑉 → ( ·𝑠𝑣) = · )
2625oveqd 7448 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑙( ·𝑠𝑣)𝑦) = (𝑙 · 𝑦))
2726eqeq2d 2746 . . . . . . . 8 (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
2822, 27rexeqbidv 3345 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦)))
2916, 28bi2anan9r 639 . . . . . 6 ((𝑣 = 𝑉𝑏 = 𝐵) → (((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))))
3013, 29syl 17 . . . . 5 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → (((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦)) ↔ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))))
3130opabbidv 5214 . . . 4 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))})
3212, 31qseq12d 42259 . . 3 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → (𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
333, 32csbied 3946 . 2 (𝑣 = 𝑉((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
34 df-prjsp 42589 . 2 ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
35 fvex 6920 . . . . 5 (Base‘𝑉) ∈ V
3635difexi 5336 . . . 4 ((Base‘𝑉) ∖ {(0g𝑉)}) ∈ V
378, 36eqeltri 2835 . . 3 𝐵 ∈ V
3837qsex 8815 . 2 (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
3933, 34, 38fvmpt 7016 1 (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  Vcvv 3478  csb 3908  cdif 3960  {csn 4631  {copab 5210  cfv 6563  (class class class)co 7431   / cqs 8743  Basecbs 17245  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17486  LVecclvec 21119  ℙ𝕣𝕠𝕛cprjsp 42588
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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-ec 8746  df-qs 8750  df-prjsp 42589
This theorem is referenced by:  prjspval2  42600  prjspnval2  42605
  Copyright terms: Public domain W3C validator