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Theorem pjfval 20244
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 6344 . . . . . . 7 (𝑤 = 𝑊 → (LSubSp‘𝑤) = (LSubSp‘𝑊))
3 pjfval.l . . . . . . 7 𝐿 = (LSubSp‘𝑊)
42, 3syl6eqr 2804 . . . . . 6 (𝑤 = 𝑊 → (LSubSp‘𝑤) = 𝐿)
5 fveq2 6344 . . . . . . . 8 (𝑤 = 𝑊 → (proj1𝑤) = (proj1𝑊))
6 pjfval.p . . . . . . . 8 𝑃 = (proj1𝑊)
75, 6syl6eqr 2804 . . . . . . 7 (𝑤 = 𝑊 → (proj1𝑤) = 𝑃)
8 eqidd 2753 . . . . . . 7 (𝑤 = 𝑊𝑥 = 𝑥)
9 fveq2 6344 . . . . . . . . 9 (𝑤 = 𝑊 → (ocv‘𝑤) = (ocv‘𝑊))
10 pjfval.o . . . . . . . . 9 = (ocv‘𝑊)
119, 10syl6eqr 2804 . . . . . . . 8 (𝑤 = 𝑊 → (ocv‘𝑤) = )
1211fveq1d 6346 . . . . . . 7 (𝑤 = 𝑊 → ((ocv‘𝑤)‘𝑥) = ( 𝑥))
137, 8, 12oveq123d 6826 . . . . . 6 (𝑤 = 𝑊 → (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥)) = (𝑥𝑃( 𝑥)))
144, 13mpteq12dv 4877 . . . . 5 (𝑤 = 𝑊 → (𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))))
15 fveq2 6344 . . . . . . . 8 (𝑤 = 𝑊 → (Base‘𝑤) = (Base‘𝑊))
16 pjfval.v . . . . . . . 8 𝑉 = (Base‘𝑊)
1715, 16syl6eqr 2804 . . . . . . 7 (𝑤 = 𝑊 → (Base‘𝑤) = 𝑉)
1817, 17oveq12d 6823 . . . . . 6 (𝑤 = 𝑊 → ((Base‘𝑤) ↑𝑚 (Base‘𝑤)) = (𝑉𝑚 𝑉))
1918xpeq2d 5288 . . . . 5 (𝑤 = 𝑊 → (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤))) = (V × (𝑉𝑚 𝑉)))
2014, 19ineq12d 3950 . . . 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 20241 . . . 4 proj = (𝑤 ∈ V ↦ ((𝑥 ∈ (LSubSp‘𝑤) ↦ (𝑥(proj1𝑤)((ocv‘𝑤)‘𝑥))) ∩ (V × ((Base‘𝑤) ↑𝑚 (Base‘𝑤)))))
22 fvex 6354 . . . . . . . 8 (LSubSp‘𝑊) ∈ V
233, 22eqeltri 2827 . . . . . . 7 𝐿 ∈ V
2423inex1 4943 . . . . . 6 (𝐿 ∩ V) ∈ V
25 ovex 6833 . . . . . . 7 (𝑉𝑚 𝑉) ∈ V
2625inex2 4944 . . . . . 6 (V ∩ (𝑉𝑚 𝑉)) ∈ V
2724, 26xpex 7119 . . . . 5 ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉))) ∈ V
28 eqid 2752 . . . . . . . 8 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
29 ovexd 6835 . . . . . . . 8 (𝑥𝐿 → (𝑥𝑃( 𝑥)) ∈ V)
3028, 29fmpti 6538 . . . . . . 7 (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V
31 fssxp 6213 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))):𝐿⟶V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V))
32 ssrin 3973 . . . . . . 7 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ⊆ (𝐿 × V) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))))
3330, 31, 32mp2b 10 . . . . . 6 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉)))
34 inxp 5402 . . . . . 6 ((𝐿 × V) ∩ (V × (𝑉𝑚 𝑉))) = ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
3533, 34sseqtri 3770 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ ((𝐿 ∩ V) × (V ∩ (𝑉𝑚 𝑉)))
3627, 35ssexi 4947 . . . 4 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ∈ V
3720, 21, 36fvmpt 6436 . . 3 (𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
38 fvprc 6338 . . . 4 𝑊 ∈ V → (proj‘𝑊) = ∅)
39 inss1 3968 . . . . 5 ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥)))
40 fvprc 6338 . . . . . . . 8 𝑊 ∈ V → (LSubSp‘𝑊) = ∅)
413, 40syl5eq 2798 . . . . . . 7 𝑊 ∈ V → 𝐿 = ∅)
4241mpteq1d 4882 . . . . . 6 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))))
43 mpt0 6174 . . . . . 6 (𝑥 ∈ ∅ ↦ (𝑥𝑃( 𝑥))) = ∅
4442, 43syl6eq 2802 . . . . 5 𝑊 ∈ V → (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅)
45 sseq0 4110 . . . . 5 ((((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) ⊆ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∧ (𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) = ∅) → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ∅)
4639, 44, 45sylancr 698 . . . 4 𝑊 ∈ V → ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))) = ∅)
4738, 46eqtr4d 2789 . . 3 𝑊 ∈ V → (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉))))
4837, 47pm2.61i 176 . 2 (proj‘𝑊) = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
491, 48eqtri 2774 1 𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉𝑚 𝑉)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  cin 3706  wss 3707  c0 4050  cmpt 4873   × cxp 5256  wf 6037  cfv 6041  (class class class)co 6805  𝑚 cmap 8015  Basecbs 16051  proj1cpj1 18242  LSubSpclss 19126  ocvcocv 20198  projcpj 20238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-ov 6808  df-pj 20241
This theorem is referenced by:  pjdm  20245  pjpm  20246  pjfval2  20247
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