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

Theorem dpjval 18655
Description: Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
Hypotheses
Ref Expression
dpjfval.1 (𝜑𝐺dom DProd 𝑆)
dpjfval.2 (𝜑 → dom 𝑆 = 𝐼)
dpjfval.p 𝑃 = (𝐺dProj𝑆)
dpjfval.q 𝑄 = (proj1𝐺)
dpjval.3 (𝜑𝑋𝐼)
Assertion
Ref Expression
dpjval (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))

Proof of Theorem dpjval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dpjfval.1 . . 3 (𝜑𝐺dom DProd 𝑆)
2 dpjfval.2 . . 3 (𝜑 → dom 𝑆 = 𝐼)
3 dpjfval.p . . 3 𝑃 = (𝐺dProj𝑆)
4 dpjfval.q . . 3 𝑄 = (proj1𝐺)
51, 2, 3, 4dpjfval 18654 . 2 (𝜑𝑃 = (𝑥𝐼 ↦ ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))))
6 simpr 479 . . . 4 ((𝜑𝑥 = 𝑋) → 𝑥 = 𝑋)
76fveq2d 6356 . . 3 ((𝜑𝑥 = 𝑋) → (𝑆𝑥) = (𝑆𝑋))
86sneqd 4333 . . . . . 6 ((𝜑𝑥 = 𝑋) → {𝑥} = {𝑋})
98difeq2d 3871 . . . . 5 ((𝜑𝑥 = 𝑋) → (𝐼 ∖ {𝑥}) = (𝐼 ∖ {𝑋}))
109reseq2d 5551 . . . 4 ((𝜑𝑥 = 𝑋) → (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝑆 ↾ (𝐼 ∖ {𝑋})))
1110oveq2d 6829 . . 3 ((𝜑𝑥 = 𝑋) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋}))))
127, 11oveq12d 6831 . 2 ((𝜑𝑥 = 𝑋) → ((𝑆𝑥)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
13 dpjval.3 . 2 (𝜑𝑋𝐼)
14 ovexd 6843 . 2 (𝜑 → ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))) ∈ V)
155, 12, 13, 14fvmptd 6450 1 (𝜑 → (𝑃𝑋) = ((𝑆𝑋)𝑄(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑋})))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  cdif 3712  {csn 4321   class class class wbr 4804  dom cdm 5266  cres 5268  cfv 6049  (class class class)co 6813  proj1cpj1 18250   DProd cdprd 18592  dProjcdpj 18593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-ixp 8075  df-dprd 18594  df-dpj 18595
This theorem is referenced by:  dpjf  18656  dpjidcl  18657  dpjlid  18660  dpjghm  18662
  Copyright terms: Public domain W3C validator