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Theorem prjspval 39131
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 6676 . . . . 5 (Base‘𝑣) ∈ V
21difexi 5223 . . . 4 ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V)
4 fveq2 6663 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 fveq2 6663 . . . . . . . . . 10 (𝑣 = 𝑉 → (0g𝑣) = (0g𝑉))
65sneqd 4569 . . . . . . . . 9 (𝑣 = 𝑉 → {(0g𝑣)} = {(0g𝑉)})
74, 6difeq12d 4097 . . . . . . . 8 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = ((Base‘𝑉) ∖ {(0g𝑉)}))
8 prjspval.b . . . . . . . 8 𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})
97, 8syl6eqr 2871 . . . . . . 7 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = 𝐵)
109eqeq2d 2829 . . . . . 6 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) ↔ 𝑏 = 𝐵))
1110biimpd 230 . . . . 5 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) → 𝑏 = 𝐵))
1211imp 407 . . . 4 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → 𝑏 = 𝐵)
1311imdistani 569 . . . . . 6 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → (𝑣 = 𝑉𝑏 = 𝐵))
14 eleq2 2898 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥𝑏𝑥𝐵))
15 eleq2 2898 . . . . . . . 8 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1614, 15anbi12d 630 . . . . . . 7 (𝑏 = 𝐵 → ((𝑥𝑏𝑦𝑏) ↔ (𝑥𝐵𝑦𝐵)))
17 fveq2 6663 . . . . . . . . . . 11 (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑉)
1917, 18syl6eqr 2871 . . . . . . . . . 10 (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
2019fveq2d 6667 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21 prjspval.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
2220, 21syl6eqr 2871 . . . . . . . 8 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23 fveq2 6663 . . . . . . . . . . 11 (𝑣 = 𝑉 → ( ·𝑠𝑣) = ( ·𝑠𝑉))
24 prjspval.x . . . . . . . . . . 11 · = ( ·𝑠𝑉)
2523, 24syl6eqr 2871 . . . . . . . . . 10 (𝑣 = 𝑉 → ( ·𝑠𝑣) = · )
2625oveqd 7162 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑙( ·𝑠𝑣)𝑦) = (𝑙 · 𝑦))
2726eqeq2d 2829 . . . . . . . 8 (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
2822, 27rexeqbidv 3400 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦)))
2916, 28bi2anan9r 636 . . . . . 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 5123 . . . 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 39002 . . 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 3916 . 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 39130 . 2 ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
35 fvex 6676 . . . . 5 (Base‘𝑉) ∈ V
3635difexi 5223 . . . 4 ((Base‘𝑉) ∖ {(0g𝑉)}) ∈ V
378, 36eqeltri 2906 . . 3 𝐵 ∈ V
3837qsex 8345 . 2 (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
3933, 34, 38fvmpt 6761 1 (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136  Vcvv 3492  csb 3880  cdif 3930  {csn 4557  {copab 5119  cfv 6348  (class class class)co 7145   / cqs 8277  Basecbs 16471  Scalarcsca 16556   ·𝑠 cvsca 16557  0gc0g 16701  LVecclvec 19803  ℙ𝕣𝕠𝕛cprjsp 39129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-ec 8280  df-qs 8284  df-prjsp 39130
This theorem is referenced by:  prjspval2  39141  prjspnval2  39145
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