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Theorem msrval 31561
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 31560 . . 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 6241 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st𝑠)) ∈ V)
7 fvexd 6241 . . . 4 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) → (2nd𝑠) ∈ V)
8 simpllr 815 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
98fveq2d 6233 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st𝑠) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
109fveq2d 6233 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11 eqid 2651 . . . . . . . . . . . . 13 (mDV‘𝑇) = (mDV‘𝑇)
12 eqid 2651 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
1311, 12, 2elmpst 31559 . . . . . . . . . . . 12 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1413simp1bi 1096 . . . . . . . . . . 11 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷))
1514simpld 474 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (mDV‘𝑇))
1615ad3antrrr 766 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17 fvex 6239 . . . . . . . . . 10 (mDV‘𝑇) ∈ V
1817ssex 4835 . . . . . . . . 9 (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
1916, 18syl 17 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ∈ V)
2013simp2bi 1097 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
2120simprd 478 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐻 ∈ Fin)
2221ad3antrrr 766 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐻 ∈ Fin)
2313simp3bi 1098 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (mEx‘𝑇))
2423ad3antrrr 766 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25 ot1stg 7224 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2619, 22, 24, 25syl3anc 1366 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2710, 26eqtrd 2685 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = 𝐷)
28 fvex 6239 . . . . . . . . . . 11 (mVars‘𝑇) ∈ V
291, 28eqeltri 2726 . . . . . . . . . 10 𝑉 ∈ V
30 imaexg 7145 . . . . . . . . . 10 (𝑉 ∈ V → (𝑉 “ ( ∪ {𝑎})) ∈ V)
3129, 30ax-mp 5 . . . . . . . . 9 (𝑉 “ ( ∪ {𝑎})) ∈ V
3231uniex 6995 . . . . . . . 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 807 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = (2nd ‘(1st𝑠)))
369fveq2d 6233 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
37 ot2ndg 7225 . . . . . . . . . . . . . . 15 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3819, 22, 24, 37syl3anc 1366 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3935, 36, 383eqtrd 2689 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = 𝐻)
40 simpr 476 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = (2nd𝑠))
418fveq2d 6233 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
42 ot3rdg 7226 . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = 𝐴)
4544sneqd 4222 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → {𝑎} = {𝐴})
4639, 45uneq12d 3801 . . . . . . . . . . . 12 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ( ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
4746imaeq2d 5501 . . . . . . . . . . 11 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
4847unieqd 4478 . . . . . . . . . 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 2703 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = 𝑍)
5134, 50sylan9eqr 2707 . . . . . . . 8 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → 𝑧 = 𝑍)
5251sqxpeqd 5175 . . . . . . 7 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
5333, 52csbied 3593 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑍 × 𝑍))
5427, 53ineq12d 3848 . . . . 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 4448 . . . 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 3593 . . 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 3593 . 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 4963 . . 3 ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
6059a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
615, 57, 58, 60fvmptd 6327 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 1523  wcel 2030  Vcvv 3231  csb 3566  cun 3605  cin 3606  wss 3607  {csn 4210  cotp 4218   cuni 4468  cmpt 4762   × cxp 5141  ccnv 5142  cima 5146  cfv 5926  1st c1st 7208  2nd c2nd 7209  Fincfn 7997  mExcmex 31490  mDVcmdv 31491  mVarscmvrs 31492  mPreStcmpst 31496  mStRedcmsr 31497
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-ot 4219  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-1st 7210  df-2nd 7211  df-mpst 31516  df-msr 31517
This theorem is referenced by:  msrf  31565  msrid  31568  elmsta  31571  mthmpps  31605
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