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Theorem prjspval 41345
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 𝐡 = ((Baseβ€˜π‘‰) βˆ– {(0gβ€˜π‘‰)})
prjspval.x Β· = ( ·𝑠 β€˜π‘‰)
prjspval.s 𝑆 = (Scalarβ€˜π‘‰)
prjspval.k 𝐾 = (Baseβ€˜π‘†)
Assertion
Ref Expression
prjspval (𝑉 ∈ LVec β†’ (β„™π•£π• π•›β€˜π‘‰) = (𝐡 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))}))
Distinct variable group:   π‘₯,𝑙,𝑦,𝑉
Allowed substitution hints:   𝐡(π‘₯,𝑦,𝑙)   𝑆(π‘₯,𝑦,𝑙)   Β· (π‘₯,𝑦,𝑙)   𝐾(π‘₯,𝑦,𝑙)

Proof of Theorem prjspval
Dummy variables 𝑏 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6905 . . . . 5 (Baseβ€˜π‘£) ∈ V
21difexi 5329 . . . 4 ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 β†’ ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) ∈ V)
4 fveq2 6892 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (Baseβ€˜π‘£) = (Baseβ€˜π‘‰))
5 fveq2 6892 . . . . . . . . . 10 (𝑣 = 𝑉 β†’ (0gβ€˜π‘£) = (0gβ€˜π‘‰))
65sneqd 4641 . . . . . . . . 9 (𝑣 = 𝑉 β†’ {(0gβ€˜π‘£)} = {(0gβ€˜π‘‰)})
74, 6difeq12d 4124 . . . . . . . 8 (𝑣 = 𝑉 β†’ ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) = ((Baseβ€˜π‘‰) βˆ– {(0gβ€˜π‘‰)}))
8 prjspval.b . . . . . . . 8 𝐡 = ((Baseβ€˜π‘‰) βˆ– {(0gβ€˜π‘‰)})
97, 8eqtr4di 2791 . . . . . . 7 (𝑣 = 𝑉 β†’ ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) = 𝐡)
109eqeq2d 2744 . . . . . 6 (𝑣 = 𝑉 β†’ (𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) ↔ 𝑏 = 𝐡))
1110biimpd 228 . . . . 5 (𝑣 = 𝑉 β†’ (𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) β†’ 𝑏 = 𝐡))
1211imp 408 . . . 4 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ 𝑏 = 𝐡)
1311imdistani 570 . . . . . 6 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ (𝑣 = 𝑉 ∧ 𝑏 = 𝐡))
14 eleq2 2823 . . . . . . . 8 (𝑏 = 𝐡 β†’ (π‘₯ ∈ 𝑏 ↔ π‘₯ ∈ 𝐡))
15 eleq2 2823 . . . . . . . 8 (𝑏 = 𝐡 β†’ (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐡))
1614, 15anbi12d 632 . . . . . . 7 (𝑏 = 𝐡 β†’ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ↔ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)))
17 fveq2 6892 . . . . . . . . . . 11 (𝑣 = 𝑉 β†’ (Scalarβ€˜π‘£) = (Scalarβ€˜π‘‰))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalarβ€˜π‘‰)
1917, 18eqtr4di 2791 . . . . . . . . . 10 (𝑣 = 𝑉 β†’ (Scalarβ€˜π‘£) = 𝑆)
2019fveq2d 6896 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (Baseβ€˜(Scalarβ€˜π‘£)) = (Baseβ€˜π‘†))
21 prjspval.k . . . . . . . . 9 𝐾 = (Baseβ€˜π‘†)
2220, 21eqtr4di 2791 . . . . . . . 8 (𝑣 = 𝑉 β†’ (Baseβ€˜(Scalarβ€˜π‘£)) = 𝐾)
23 fveq2 6892 . . . . . . . . . . 11 (𝑣 = 𝑉 β†’ ( ·𝑠 β€˜π‘£) = ( ·𝑠 β€˜π‘‰))
24 prjspval.x . . . . . . . . . . 11 Β· = ( ·𝑠 β€˜π‘‰)
2523, 24eqtr4di 2791 . . . . . . . . . 10 (𝑣 = 𝑉 β†’ ( ·𝑠 β€˜π‘£) = Β· )
2625oveqd 7426 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (𝑙( ·𝑠 β€˜π‘£)𝑦) = (𝑙 Β· 𝑦))
2726eqeq2d 2744 . . . . . . . 8 (𝑣 = 𝑉 β†’ (π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦) ↔ π‘₯ = (𝑙 Β· 𝑦)))
2822, 27rexeqbidv 3344 . . . . . . 7 (𝑣 = 𝑉 β†’ (βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦) ↔ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦)))
2916, 28bi2anan9r 639 . . . . . 6 ((𝑣 = 𝑉 ∧ 𝑏 = 𝐡) β†’ (((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦)) ↔ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))))
3013, 29syl 17 . . . . 5 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ (((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦)) ↔ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))))
3130opabbidv 5215 . . . 4 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))} = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))})
3212, 31qseq12d 41061 . . 3 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ (𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}) = (𝐡 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))}))
333, 32csbied 3932 . 2 (𝑣 = 𝑉 β†’ ⦋((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) / π‘β¦Œ(𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}) = (𝐡 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))}))
34 df-prjsp 41344 . 2 ℙ𝕣𝕠𝕛 = (𝑣 ∈ LVec ↦ ⦋((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) / π‘β¦Œ(𝑏 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ∧ βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦))}))
35 fvex 6905 . . . . 5 (Baseβ€˜π‘‰) ∈ V
3635difexi 5329 . . . 4 ((Baseβ€˜π‘‰) βˆ– {(0gβ€˜π‘‰)}) ∈ V
378, 36eqeltri 2830 . . 3 𝐡 ∈ V
3837qsex 8770 . 2 (𝐡 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))}) ∈ V
3933, 34, 38fvmpt 6999 1 (𝑉 ∈ LVec β†’ (β„™π•£π• π•›β€˜π‘‰) = (𝐡 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475  β¦‹csb 3894   βˆ– cdif 3946  {csn 4629  {copab 5211  β€˜cfv 6544  (class class class)co 7409   / cqs 8702  Basecbs 17144  Scalarcsca 17200   ·𝑠 cvsca 17201  0gc0g 17385  LVecclvec 20713  β„™π•£π• π•›cprjsp 41343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-ec 8705  df-qs 8709  df-prjsp 41344
This theorem is referenced by:  prjspval2  41355  prjspnval2  41360
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