Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  pjfval Structured version   Visualization version   GIF version

Theorem pjfval 19869
 Description: The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
Hypotheses
Ref Expression
pjfval.v 𝑉 = (Base‘𝑊)
pjfval.l 𝐿 = (LSubSp‘𝑊)
pjfval.o = (ocv‘𝑊)
pjfval.p 𝑃 = (proj1𝑊)
pjfval.k 𝐾 = (proj‘𝑊)
Assertion
Ref Expression
pjfval 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
Distinct variable groups:   𝑥,   𝑥,𝐿   𝑥,𝑃   𝑥,𝑉   𝑥,𝑊
Allowed substitution hint:   𝐾(𝑥)

Proof of Theorem pjfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 pjfval.k . 2 𝐾 = (proj‘𝑊)
2 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3 pjfval.l . . . . . . 7 𝐿 = (LSubSp‘𝑊)
42, 3syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → (proj1𝑤) = (proj1𝑊))
6 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
75, 6syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (proj1𝑤) = 𝑃)
8 eqidd 2611 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6103 . . . . . . . . 9 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10 pjfval.o . . . . . . . . 9 = (ocv‘𝑊)
119, 10syl6eqr 2662 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = )
1211fveq1d 6105 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( 𝑥))
137, 8, 12oveq123d 6570 . . . . . 6 (𝑤 = 𝑊 → (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( 𝑥)))
144, 13mpteq12dv 4663 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))))
15 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16 pjfval.v . . . . . . . 8 𝑉 = (Base‘𝑊)
1715, 16syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
1817, 17oveq12d 6567 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ↑𝑚 (Base‘𝑤)) = (𝑉𝑚 𝑉))
1918xpeq2d 5063 . . . . 5 (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤))) = (V × (𝑉𝑚 𝑉)))
2014, 19ineq12d 3777 . . . 4 (𝑤 = 𝑊 → ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤)))) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
21 df-pj 19866 . . . 4 proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤)))))
22 fvex 6113 . . . . . . . 8 (LSubSp‘𝑊) ∈ V
233, 22eqeltri 2684 . . . . . . 7 𝐿 ∈ V
2423inex1 4727 . . . . . 6 (𝐿 ∩ V) ∈ V
25 ovex 6577 . . . . . . 7 (𝑉𝑚 𝑉) ∈ V
2625inex2 4728 . . . . . 6 (V ∩ (𝑉𝑚 𝑉)) ∈ V
2724, 26xpex 6860 . . . . 5 ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉))) ∈ V
28 eqid 2610 . . . . . . . 8 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
29 ovex 6577 . . . . . . . . 9 (𝑥𝑃( 𝑥)) ∈ V
3029a1i 11 . . . . . . . 8 (𝑥𝐿 → (𝑥𝑃( 𝑥)) ∈ V)
3128, 30fmpti 6291 . . . . . . 7 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V
32 fssxp 5973 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V))
33 ssrin 3800 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))))
3431, 32, 33mp2b 10 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
35 inxp 5176 . . . . . 6 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
3634, 35sseqtri 3600 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
3727, 36ssexi 4731 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ∈ V
3820, 21, 37fvmpt 6191 . . 3 (𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
39 fvprc 6097 . . . 4 𝑊 ∈ V → (proj‘𝑊) = ∅)
40 inss1 3795 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
41 fvprc 6097 . . . . . . . 8 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
423, 41syl5eq 2656 . . . . . . 7 𝑊 ∈ V → 𝐿 = ∅)
4342mpteq1d 4666 . . . . . 6 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))))
44 mpt0 5934 . . . . . 6 (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))) = ∅
4543, 44syl6eq 2660 . . . . 5 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅)
46 sseq0 3927 . . . . 5 ((((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∧ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ∅)
4740, 45, 46sylancr 694 . . . 4 𝑊 ∈ V → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ∅)
4839, 47eqtr4d 2647 . . 3 𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
4938, 48pm2.61i 175 . 2 (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
501, 49eqtri 2632 1 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874   ↦ cmpt 4643   × cxp 5036  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  Basecbs 15695  proj1cpj1 17873  LSubSpclss 18753  ocvcocv 19823  projcpj 19863 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-pj 19866 This theorem is referenced by:  pjdm  19870  pjpm  19871  pjfval2  19872
 Copyright terms: Public domain W3C validator