Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mrsubval Structured version   Visualization version   GIF version

Theorem mrsubval 31532
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c 𝐶 = (mCN‘𝑇)
mrsubffval.v 𝑉 = (mVR‘𝑇)
mrsubffval.r 𝑅 = (mREx‘𝑇)
mrsubffval.s 𝑆 = (mRSubst‘𝑇)
mrsubffval.g 𝐺 = (freeMnd‘(𝐶𝑉))
Assertion
Ref Expression
mrsubval ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
Distinct variable groups:   𝑣,𝐴   𝑣,𝐶   𝑣,𝐹   𝑣,𝑅   𝑣,𝑋   𝑣,𝑇   𝑣,𝑉
Allowed substitution hints:   𝑆(𝑣)   𝐺(𝑣)

Proof of Theorem mrsubval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.c . . . 4 𝐶 = (mCN‘𝑇)
2 mrsubffval.v . . . 4 𝑉 = (mVR‘𝑇)
3 mrsubffval.r . . . 4 𝑅 = (mREx‘𝑇)
4 mrsubffval.s . . . 4 𝑆 = (mRSubst‘𝑇)
5 mrsubffval.g . . . 4 𝐺 = (freeMnd‘(𝐶𝑉))
61, 2, 3, 4, 5mrsubfval 31531 . . 3 ((𝐹:𝐴𝑅𝐴𝑉) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
763adant3 1101 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → (𝑆𝐹) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
8 simpr 476 . . . 4 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋)
98coeq2d 5317 . . 3 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))
109oveq2d 6706 . 2 (((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) ∧ 𝑒 = 𝑋) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
11 simp3 1083 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → 𝑋𝑅)
12 ovexd 6720 . 2 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ V)
137, 10, 11, 12fvmptd 6327 1 ((𝐹:𝐴𝑅𝐴𝑉𝑋𝑅) → ((𝑆𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝐴, (𝐹𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  Vcvv 3231  cun 3605  wss 3607  ifcif 4119  cmpt 4762  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  ⟨“cs1 13326   Σg cgsu 16148  freeMndcfrmd 17431  mCNcmcn 31483  mVRcmvar 31484  mRExcmrex 31489  mRSubstcmrsub 31493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-pm 7902  df-mrsub 31513
This theorem is referenced by:  mrsubcv  31533  mrsub0  31539  mrsubccat  31541
  Copyright terms: Public domain W3C validator