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Theorem prjspval 40927
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 6855 . . . . 5 (Base‘𝑣) ∈ V
21difexi 5285 . . . 4 ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) ∈ V)
4 fveq2 6842 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘𝑣) = (Base‘𝑉))
5 fveq2 6842 . . . . . . . . . 10 (𝑣 = 𝑉 → (0g𝑣) = (0g𝑉))
65sneqd 4598 . . . . . . . . 9 (𝑣 = 𝑉 → {(0g𝑣)} = {(0g𝑉)})
74, 6difeq12d 4083 . . . . . . . 8 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = ((Base‘𝑉) ∖ {(0g𝑉)}))
8 prjspval.b . . . . . . . 8 𝐵 = ((Base‘𝑉) ∖ {(0g𝑉)})
97, 8eqtr4di 2794 . . . . . . 7 (𝑣 = 𝑉 → ((Base‘𝑣) ∖ {(0g𝑣)}) = 𝐵)
109eqeq2d 2747 . . . . . 6 (𝑣 = 𝑉 → (𝑏 = ((Base‘𝑣) ∖ {(0g𝑣)}) ↔ 𝑏 = 𝐵))
1110biimpd 228 . . . . 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 2826 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥𝑏𝑥𝐵))
15 eleq2 2826 . . . . . . . 8 (𝑏 = 𝐵 → (𝑦𝑏𝑦𝐵))
1614, 15anbi12d 631 . . . . . . 7 (𝑏 = 𝐵 → ((𝑥𝑏𝑦𝑏) ↔ (𝑥𝐵𝑦𝐵)))
17 fveq2 6842 . . . . . . . . . . 11 (𝑣 = 𝑉 → (Scalar‘𝑣) = (Scalar‘𝑉))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalar‘𝑉)
1917, 18eqtr4di 2794 . . . . . . . . . 10 (𝑣 = 𝑉 → (Scalar‘𝑣) = 𝑆)
2019fveq2d 6846 . . . . . . . . 9 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = (Base‘𝑆))
21 prjspval.k . . . . . . . . 9 𝐾 = (Base‘𝑆)
2220, 21eqtr4di 2794 . . . . . . . 8 (𝑣 = 𝑉 → (Base‘(Scalar‘𝑣)) = 𝐾)
23 fveq2 6842 . . . . . . . . . . 11 (𝑣 = 𝑉 → ( ·𝑠𝑣) = ( ·𝑠𝑉))
24 prjspval.x . . . . . . . . . . 11 · = ( ·𝑠𝑉)
2523, 24eqtr4di 2794 . . . . . . . . . 10 (𝑣 = 𝑉 → ( ·𝑠𝑣) = · )
2625oveqd 7374 . . . . . . . . 9 (𝑣 = 𝑉 → (𝑙( ·𝑠𝑣)𝑦) = (𝑙 · 𝑦))
2726eqeq2d 2747 . . . . . . . 8 (𝑣 = 𝑉 → (𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ 𝑥 = (𝑙 · 𝑦)))
2822, 27rexeqbidv 3320 . . . . . . 7 (𝑣 = 𝑉 → (∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦) ↔ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦)))
2916, 28bi2anan9r 638 . . . . . 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 5171 . . . 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 40662 . . 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 3893 . 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 40926 . 2 ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ((Base‘𝑣) ∖ {(0g𝑣)}) / 𝑏(𝑏 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝑏𝑦𝑏) ∧ ∃𝑙 ∈ (Base‘(Scalar‘𝑣))𝑥 = (𝑙( ·𝑠𝑣)𝑦))}))
35 fvex 6855 . . . . 5 (Base‘𝑉) ∈ V
3635difexi 5285 . . . 4 ((Base‘𝑉) ∖ {(0g𝑉)}) ∈ V
378, 36eqeltri 2834 . . 3 𝐵 ∈ V
3837qsex 8715 . 2 (𝐵 / {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}) ∈ V
3933, 34, 38fvmpt 6948 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 396   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  csb 3855  cdif 3907  {csn 4586  {copab 5167  cfv 6496  (class class class)co 7357   / cqs 8647  Basecbs 17083  Scalarcsca 17136   ·𝑠 cvsca 17137  0gc0g 17321  LVecclvec 20563  ℙ𝕣𝕠𝕛cprjsp 40925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-ec 8650  df-qs 8654  df-prjsp 40926
This theorem is referenced by:  prjspval2  40937  prjspnval2  40942
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