HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  pjhval Structured version   Visualization version   GIF version

Theorem pjhval 31426
Description: Value of a projection. (Contributed by NM, 23-Oct-1999.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pjhval ((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐻   𝑥,𝐴,𝑦

Proof of Theorem pjhval
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 pjhfval 31425 . . 3 (𝐻C → (proj𝐻) = (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦))))
21fveq1d 6909 . 2 (𝐻C → ((proj𝐻)‘𝐴) = ((𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))‘𝐴))
3 eqeq1 2739 . . . . 5 (𝑧 = 𝐴 → (𝑧 = (𝑥 + 𝑦) ↔ 𝐴 = (𝑥 + 𝑦)))
43rexbidv 3177 . . . 4 (𝑧 = 𝐴 → (∃𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦) ↔ ∃𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
54riotabidv 7390 . . 3 (𝑧 = 𝐴 → (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
6 eqid 2735 . . 3 (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦))) = (𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))
7 riotaex 7392 . . 3 (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)) ∈ V
85, 6, 7fvmpt 7016 . 2 (𝐴 ∈ ℋ → ((𝑧 ∈ ℋ ↦ (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝑧 = (𝑥 + 𝑦)))‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
92, 8sylan9eq 2795 1 ((𝐻C𝐴 ∈ ℋ) → ((proj𝐻)‘𝐴) = (𝑥𝐻𝑦 ∈ (⊥‘𝐻)𝐴 = (𝑥 + 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  cmpt 5231  cfv 6563  crio 7387  (class class class)co 7431  chba 30948   + cva 30949   C cch 30958  cort 30959  projcpjh 30966
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-hilex 31028
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-pjh 31424
This theorem is referenced by:  pjpreeq  31427
  Copyright terms: Public domain W3C validator