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 41346
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 6904 . . . . 5 (Baseβ€˜π‘£) ∈ V
21difexi 5328 . . . 4 ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) ∈ V
32a1i 11 . . 3 (𝑣 = 𝑉 β†’ ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) ∈ V)
4 fveq2 6891 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (Baseβ€˜π‘£) = (Baseβ€˜π‘‰))
5 fveq2 6891 . . . . . . . . . 10 (𝑣 = 𝑉 β†’ (0gβ€˜π‘£) = (0gβ€˜π‘‰))
65sneqd 4640 . . . . . . . . 9 (𝑣 = 𝑉 β†’ {(0gβ€˜π‘£)} = {(0gβ€˜π‘‰)})
74, 6difeq12d 4123 . . . . . . . 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 2790 . . . . . . 7 (𝑣 = 𝑉 β†’ ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)}) = 𝐡)
109eqeq2d 2743 . . . . . 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 407 . . . 4 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ 𝑏 = 𝐡)
1311imdistani 569 . . . . . 6 ((𝑣 = 𝑉 ∧ 𝑏 = ((Baseβ€˜π‘£) βˆ– {(0gβ€˜π‘£)})) β†’ (𝑣 = 𝑉 ∧ 𝑏 = 𝐡))
14 eleq2 2822 . . . . . . . 8 (𝑏 = 𝐡 β†’ (π‘₯ ∈ 𝑏 ↔ π‘₯ ∈ 𝐡))
15 eleq2 2822 . . . . . . . 8 (𝑏 = 𝐡 β†’ (𝑦 ∈ 𝑏 ↔ 𝑦 ∈ 𝐡))
1614, 15anbi12d 631 . . . . . . 7 (𝑏 = 𝐡 β†’ ((π‘₯ ∈ 𝑏 ∧ 𝑦 ∈ 𝑏) ↔ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)))
17 fveq2 6891 . . . . . . . . . . 11 (𝑣 = 𝑉 β†’ (Scalarβ€˜π‘£) = (Scalarβ€˜π‘‰))
18 prjspval.s . . . . . . . . . . 11 𝑆 = (Scalarβ€˜π‘‰)
1917, 18eqtr4di 2790 . . . . . . . . . 10 (𝑣 = 𝑉 β†’ (Scalarβ€˜π‘£) = 𝑆)
2019fveq2d 6895 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (Baseβ€˜(Scalarβ€˜π‘£)) = (Baseβ€˜π‘†))
21 prjspval.k . . . . . . . . 9 𝐾 = (Baseβ€˜π‘†)
2220, 21eqtr4di 2790 . . . . . . . 8 (𝑣 = 𝑉 β†’ (Baseβ€˜(Scalarβ€˜π‘£)) = 𝐾)
23 fveq2 6891 . . . . . . . . . . 11 (𝑣 = 𝑉 β†’ ( ·𝑠 β€˜π‘£) = ( ·𝑠 β€˜π‘‰))
24 prjspval.x . . . . . . . . . . 11 Β· = ( ·𝑠 β€˜π‘‰)
2523, 24eqtr4di 2790 . . . . . . . . . 10 (𝑣 = 𝑉 β†’ ( ·𝑠 β€˜π‘£) = Β· )
2625oveqd 7425 . . . . . . . . 9 (𝑣 = 𝑉 β†’ (𝑙( ·𝑠 β€˜π‘£)𝑦) = (𝑙 Β· 𝑦))
2726eqeq2d 2743 . . . . . . . 8 (𝑣 = 𝑉 β†’ (π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦) ↔ π‘₯ = (𝑙 Β· 𝑦)))
2822, 27rexeqbidv 3343 . . . . . . 7 (𝑣 = 𝑉 β†’ (βˆƒπ‘™ ∈ (Baseβ€˜(Scalarβ€˜π‘£))π‘₯ = (𝑙( ·𝑠 β€˜π‘£)𝑦) ↔ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦)))
2916, 28bi2anan9r 638 . . . . . 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 5214 . . . 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 41063 . . 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 3931 . 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 41345 . 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 6904 . . . . 5 (Baseβ€˜π‘‰) ∈ V
3635difexi 5328 . . . 4 ((Baseβ€˜π‘‰) βˆ– {(0gβ€˜π‘‰)}) ∈ V
378, 36eqeltri 2829 . . 3 𝐡 ∈ V
3837qsex 8769 . 2 (𝐡 / {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ∧ βˆƒπ‘™ ∈ 𝐾 π‘₯ = (𝑙 Β· 𝑦))}) ∈ V
3933, 34, 38fvmpt 6998 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474  β¦‹csb 3893   βˆ– cdif 3945  {csn 4628  {copab 5210  β€˜cfv 6543  (class class class)co 7408   / cqs 8701  Basecbs 17143  Scalarcsca 17199   ·𝑠 cvsca 17200  0gc0g 17384  LVecclvec 20712  β„™π•£π• π•›cprjsp 41344
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-ec 8704  df-qs 8708  df-prjsp 41345
This theorem is referenced by:  prjspval2  41356  prjspnval2  41361
  Copyright terms: Public domain W3C validator