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Theorem msrval 31713
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 31712 . . 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 6352 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st𝑠)) ∈ V)
7 fvexd 6352 . . . 4 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) → (2nd𝑠) ∈ V)
8 simpllr 817 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
98fveq2d 6344 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st𝑠) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
109fveq2d 6344 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11 eqid 2748 . . . . . . . . . . . . 13 (mDV‘𝑇) = (mDV‘𝑇)
12 eqid 2748 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
1311, 12, 2elmpst 31711 . . . . . . . . . . . 12 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1413simp1bi 1137 . . . . . . . . . . 11 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷))
1514simpld 477 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (mDV‘𝑇))
1615ad3antrrr 768 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17 fvex 6350 . . . . . . . . . 10 (mDV‘𝑇) ∈ V
1817ssex 4942 . . . . . . . . 9 (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
1916, 18syl 17 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ∈ V)
2013simp2bi 1138 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
2120simprd 482 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐻 ∈ Fin)
2221ad3antrrr 768 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐻 ∈ Fin)
2313simp3bi 1139 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (mEx‘𝑇))
2423ad3antrrr 768 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25 ot1stg 7335 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2619, 22, 24, 25syl3anc 1463 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2710, 26eqtrd 2782 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = 𝐷)
28 fvex 6350 . . . . . . . . . . 11 (mVars‘𝑇) ∈ V
291, 28eqeltri 2823 . . . . . . . . . 10 𝑉 ∈ V
30 imaexg 7256 . . . . . . . . . 10 (𝑉 ∈ V → (𝑉 “ ( ∪ {𝑎})) ∈ V)
3129, 30ax-mp 5 . . . . . . . . 9 (𝑉 “ ( ∪ {𝑎})) ∈ V
3231uniex 7106 . . . . . . . 8 (𝑉 “ ( ∪ {𝑎})) ∈ V
3332a1i 11 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) ∈ V)
34 id 22 . . . . . . . . 9 (𝑧 = (𝑉 “ ( ∪ {𝑎})) → 𝑧 = (𝑉 “ ( ∪ {𝑎})))
35 simplr 809 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = (2nd ‘(1st𝑠)))
369fveq2d 6344 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
37 ot2ndg 7336 . . . . . . . . . . . . . . 15 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3819, 22, 24, 37syl3anc 1463 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3935, 36, 383eqtrd 2786 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = 𝐻)
40 simpr 479 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = (2nd𝑠))
418fveq2d 6344 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
42 ot3rdg 7337 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (mEx‘𝑇) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
4324, 42syl 17 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘⟨𝐷, 𝐻, 𝐴⟩) = 𝐴)
4440, 41, 433eqtrd 2786 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = 𝐴)
4544sneqd 4321 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → {𝑎} = {𝐴})
4639, 45uneq12d 3899 . . . . . . . . . . . 12 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ( ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
4746imaeq2d 5612 . . . . . . . . . . 11 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
4847unieqd 4586 . . . . . . . . . 10 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
49 msrval.z . . . . . . . . . 10 𝑍 = (𝑉 “ (𝐻 ∪ {𝐴}))
5048, 49syl6eqr 2800 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = 𝑍)
5134, 50sylan9eqr 2804 . . . . . . . 8 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → 𝑧 = 𝑍)
5251sqxpeqd 5286 . . . . . . 7 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
5333, 52csbied 3689 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑍 × 𝑍))
5427, 53ineq12d 3946 . . . . 5 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ((1st ‘(1st𝑠)) ∩ (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧)) = (𝐷 ∩ (𝑍 × 𝑍)))
5554, 39, 44oteq123d 4556 . . . 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⟩)
567, 55csbied 3689 . . 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⟩)
576, 56csbied 3689 . 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⟩)
58 id 22 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
59 otex 5070 . . 3 ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
6059a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
615, 57, 58, 60fvmptd 6438 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 383   = wceq 1620  wcel 2127  Vcvv 3328  csb 3662  cun 3701  cin 3702  wss 3703  {csn 4309  cotp 4317   cuni 4576  cmpt 4869   × cxp 5252  ccnv 5253  cima 5257  cfv 6037  1st c1st 7319  2nd c2nd 7320  Fincfn 8109  mExcmex 31642  mDVcmdv 31643  mVarscmvrs 31644  mPreStcmpst 31648  mStRedcmsr 31649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-ot 4318  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-1st 7321  df-2nd 7322  df-mpst 31668  df-msr 31669
This theorem is referenced by:  msrf  31717  msrid  31720  elmsta  31723  mthmpps  31757
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