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

Theorem msrval 32669
Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msrfval.v 𝑉 = (mVars‘𝑇)
msrfval.p 𝑃 = (mPreSt‘𝑇)
msrfval.r 𝑅 = (mStRed‘𝑇)
msrval.z 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
Assertion
Ref Expression
msrval (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)

Proof of Theorem msrval
Dummy variables 𝑎 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msrfval.v . . . 4 𝑉 = (mVars‘𝑇)
2 msrfval.p . . . 4 𝑃 = (mPreSt‘𝑇)
3 msrfval.r . . . 4 𝑅 = (mStRed‘𝑇)
41, 2, 3msrfval 32668 . . 3 𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩)
54a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑅 = (𝑠𝑃(2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩))
6 fvexd 6682 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st𝑠)) ∈ V)
7 fvexd 6682 . . . 4 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) → (2nd𝑠) ∈ V)
8 simpllr 772 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
98fveq2d 6671 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st𝑠) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
109fveq2d 6671 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11 eqid 2826 . . . . . . . . . . . . 13 (mDV‘𝑇) = (mDV‘𝑇)
12 eqid 2826 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
1311, 12, 2elmpst 32667 . . . . . . . . . . . 12 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1413simp1bi 1139 . . . . . . . . . . 11 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷))
1514simpld 495 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (mDV‘𝑇))
1615ad3antrrr 726 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17 fvex 6680 . . . . . . . . . 10 (mDV‘𝑇) ∈ V
1817ssex 5222 . . . . . . . . 9 (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
1916, 18syl 17 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ∈ V)
2013simp2bi 1140 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
2120simprd 496 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐻 ∈ Fin)
2221ad3antrrr 726 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐻 ∈ Fin)
2313simp3bi 1141 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (mEx‘𝑇))
2423ad3antrrr 726 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25 ot1stg 7694 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2619, 22, 24, 25syl3anc 1365 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2710, 26eqtrd 2861 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = 𝐷)
281fvexi 6681 . . . . . . . . . 10 𝑉 ∈ V
29 imaexg 7608 . . . . . . . . . 10 (𝑉 ∈ V → (𝑉 “ ( ∪ {𝑎})) ∈ V)
3028, 29ax-mp 5 . . . . . . . . 9 (𝑉 “ ( ∪ {𝑎})) ∈ V
3130uniex 7455 . . . . . . . 8 (𝑉 “ ( ∪ {𝑎})) ∈ V
3231a1i 11 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) ∈ V)
33 id 22 . . . . . . . . 9 (𝑧 = (𝑉 “ ( ∪ {𝑎})) → 𝑧 = (𝑉 “ ( ∪ {𝑎})))
34 simplr 765 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = (2nd ‘(1st𝑠)))
359fveq2d 6671 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
36 ot2ndg 7695 . . . . . . . . . . . . . . 15 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3719, 22, 24, 36syl3anc 1365 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3834, 35, 373eqtrd 2865 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = 𝐻)
39 simpr 485 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = (2nd𝑠))
408fveq2d 6671 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
41 ot3rdg 7696 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (mEx‘𝑇) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
4224, 41syl 17 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
4339, 40, 423eqtrd 2865 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = 𝐴)
4443sneqd 4576 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → {𝑎} = {𝐴})
4538, 44uneq12d 4144 . . . . . . . . . . . 12 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ( ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
4645imaeq2d 5927 . . . . . . . . . . 11 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
4746unieqd 4847 . . . . . . . . . 10 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
48 msrval.z . . . . . . . . . 10 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
4947, 48syl6eqr 2879 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = 𝑍)
5033, 49sylan9eqr 2883 . . . . . . . 8 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → 𝑧 = 𝑍)
5150sqxpeqd 5586 . . . . . . 7 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
5232, 51csbied 3923 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑍 × 𝑍))
5327, 52ineq12d 4194 . . . . 5 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)) = (𝐷 ∩ (𝑍 × 𝑍)))
5453, 38, 43oteq123d 4817 . . . 4 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
557, 54csbied 3923 . . 3 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) → (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
566, 55csbied 3923 . 2 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st𝑠)) / (2nd𝑠) / 𝑎⟨((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)), , 𝑎⟩ = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
57 id 22 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
58 otex 5354 . . 3 ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
5958a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
605, 56, 57, 59fvmptd 6771 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝑅‘⟨𝐷, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  Vcvv 3500  csb 3887  cun 3938  cin 3939  wss 3940  {csn 4564  cotp 4572   cuni 4837  cmpt 5143   × cxp 5552  ccnv 5553  cima 5557  cfv 6352  1st c1st 7678  2nd c2nd 7679  Fincfn 8498  mExcmex 32598  mDVcmdv 32599  mVarscmvrs 32600  mPreStcmpst 32604  mStRedcmsr 32605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-ot 4573  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-1st 7680  df-2nd 7681  df-mpst 32624  df-msr 32625
This theorem is referenced by:  msrf  32673  msrid  32676  elmsta  32679  mthmpps  32713
  Copyright terms: Public domain W3C validator