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Theorem prjspval 38666
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 6512 . . . . 5 (Base‘𝑣) ∈ V
21difexi 5088 . . . 4 ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V)
4 fveq2 6499 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 fveq2 6499 . . . . . . . . . 10 (𝑣 = 𝑉 → (0g𝑣) = (0g𝑉))
65sneqd 4453 . . . . . . . . 9 (𝑣 = 𝑉 → {(0g𝑣)} = {(0g𝑉)})
74, 6difeq12d 3990 . . . . . . . 8 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = ((Base‘𝑉) ∖ {(0g𝑉)}))
8 prjspval.b . . . . . . . 8 𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})
97, 8syl6eqr 2832 . . . . . . 7 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = 𝐵)
109eqeq2d 2788 . . . . . 6 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) ↔ 𝑏 = 𝐵))
1110biimpd 221 . . . . 5 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) → 𝑏 = 𝐵))
1211imp 398 . . . 4 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → 𝑏 = 𝐵)
1311imdistani 561 . . . . . 6 ((𝑣 = 𝑉𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)})) → (𝑣 = 𝑉𝑏 = 𝐵))
14 eleq2 2854 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥𝑏𝑥𝐵))
15 eleq2 2854 . . . . . . . 8 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1614, 15anbi12d 621 . . . . . . 7 (𝑏 = 𝐵 → ((𝑥𝑏𝑦𝑏) ↔ (𝑥𝐵𝑦𝐵)))
17 fveq2 6499 . . . . . . . . . . 11 (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑉)
1917, 18syl6eqr 2832 . . . . . . . . . 10 (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
2019fveq2d 6503 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21 prjspval.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
2220, 21syl6eqr 2832 . . . . . . . 8 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23 fveq2 6499 . . . . . . . . . . 11 (𝑣 = 𝑉 → ( ·𝑠𝑣) = ( ·𝑠𝑉))
24 prjspval.x . . . . . . . . . . 11 · = ( ·𝑠𝑉)
2523, 24syl6eqr 2832 . . . . . . . . . 10 (𝑣 = 𝑉 → ( ·𝑠𝑣) = · )
2625oveqd 6993 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑙( ·𝑠𝑣)𝑦) = (𝑙 · 𝑦))
2726eqeq2d 2788 . . . . . . . 8 (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
2822, 27rexeqbidv 3342 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦)))
2916, 28bi2anan9r 627 . . . . . 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 4995 . . . 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 38575 . . 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 3815 . 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 38665 . 2 ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
35 fvex 6512 . . . . 5 (Base‘𝑉) ∈ V
3635difexi 5088 . . . 4 ((Base‘𝑉) ∖ {(0g𝑉)}) ∈ V
378, 36eqeltri 2862 . . 3 𝐵 ∈ V
3837qsex 8156 . 2 (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
3933, 34, 38fvmpt 6595 1 (𝑉 ∈ LVec → (ℙ𝕣𝕠𝕛‘𝑉) = (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wrex 3089  Vcvv 3415  csb 3786  cdif 3826  {csn 4441  {copab 4991  cfv 6188  (class class class)co 6976   / cqs 8088  Basecbs 16339  Scalarcsca 16424   ·𝑠 cvsca 16425  0gc0g 16569  LVecclvec 19596  ℙ𝕣𝕠𝕛cprjsp 38664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-ov 6979  df-ec 8091  df-qs 8095  df-prjsp 38665
This theorem is referenced by:  prjspval2  38676  prjspnval2  38680
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