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 42162
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 6909 . . . . 5 (Base‘𝑣) ∈ V
21difexi 5331 . . . 4 ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V)
4 fveq2 6896 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 fveq2 6896 . . . . . . . . . 10 (𝑣 = 𝑉 → (0g𝑣) = (0g𝑉))
65sneqd 4642 . . . . . . . . 9 (𝑣 = 𝑉 → {(0g𝑣)} = {(0g𝑉)})
74, 6difeq12d 4119 . . . . . . . 8 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = ((Base‘𝑉) ∖ {(0g𝑉)}))
8 prjspval.b . . . . . . . 8 𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})
97, 8eqtr4di 2783 . . . . . . 7 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = 𝐵)
109eqeq2d 2736 . . . . . 6 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) ↔ 𝑏 = 𝐵))
1110biimpd 228 . . . . 5 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) → 𝑏 = 𝐵))
1211imp 405 . . . 4 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → 𝑏 = 𝐵)
1311imdistani 567 . . . . . 6 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → (𝑣 = 𝑉𝑏 = 𝐵))
14 eleq2 2814 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥𝑏𝑥𝐵))
15 eleq2 2814 . . . . . . . 8 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1614, 15anbi12d 630 . . . . . . 7 (𝑏 = 𝐵 → ((𝑥𝑏𝑦𝑏) ↔ (𝑥𝐵𝑦𝐵)))
17 fveq2 6896 . . . . . . . . . . 11 (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑉)
1917, 18eqtr4di 2783 . . . . . . . . . 10 (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
2019fveq2d 6900 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21 prjspval.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
2220, 21eqtr4di 2783 . . . . . . . 8 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23 fveq2 6896 . . . . . . . . . . 11 (𝑣 = 𝑉 → ( ·𝑠𝑣) = ( ·𝑠𝑉))
24 prjspval.x . . . . . . . . . . 11 · = ( ·𝑠𝑉)
2523, 24eqtr4di 2783 . . . . . . . . . 10 (𝑣 = 𝑉 → ( ·𝑠𝑣) = · )
2625oveqd 7436 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑙( ·𝑠𝑣)𝑦) = (𝑙 · 𝑦))
2726eqeq2d 2736 . . . . . . . 8 (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
2822, 27rexeqbidv 3330 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦)))
2916, 28bi2anan9r 637 . . . . . 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 5215 . . . 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 41863 . . 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 3927 . 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 42161 . 2 ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
35 fvex 6909 . . . . 5 (Base‘𝑉) ∈ V
3635difexi 5331 . . . 4 ((Base‘𝑉) ∖ {(0g𝑉)}) ∈ V
378, 36eqeltri 2821 . . 3 𝐵 ∈ V
3837qsex 8795 . 2 (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
3933, 34, 38fvmpt 7004 1 (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wrex 3059  Vcvv 3461  csb 3889  cdif 3941  {csn 4630  {copab 5211  cfv 6549  (class class class)co 7419   / cqs 8724  Basecbs 17183  Scalarcsca 17239   ·𝑠 cvsca 17240  0gc0g 17424  LVecclvec 20999  ℙ𝕣𝕠𝕛cprjsp 42160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-ec 8727  df-qs 8731  df-prjsp 42161
This theorem is referenced by:  prjspval2  42172  prjspnval2  42177
  Copyright terms: Public domain W3C validator