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Theorem msrval 35543
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 35542 . . 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 6921 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st𝑠)) ∈ V)
7 fvexd 6921 . . . 4 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) → (2nd𝑠) ∈ V)
8 simpllr 776 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
98fveq2d 6910 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st𝑠) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
109fveq2d 6910 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11 eqid 2737 . . . . . . . . . . . . 13 (mDV‘𝑇) = (mDV‘𝑇)
12 eqid 2737 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
1311, 12, 2elmpst 35541 . . . . . . . . . . . 12 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1413simp1bi 1146 . . . . . . . . . . 11 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷))
1514simpld 494 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐷 ⊆ (mDV‘𝑇))
1615ad3antrrr 730 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ⊆ (mDV‘𝑇))
17 fvex 6919 . . . . . . . . . 10 (mDV‘𝑇) ∈ V
1817ssex 5321 . . . . . . . . 9 (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
1916, 18syl 17 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ∈ V)
2013simp2bi 1147 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
2120simprd 495 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐻 ∈ Fin)
2221ad3antrrr 730 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐻 ∈ Fin)
2313simp3bi 1148 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (mEx‘𝑇))
2423ad3antrrr 730 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐴 ∈ (mEx‘𝑇))
25 ot1stg 8028 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2619, 22, 24, 25syl3anc 1373 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2710, 26eqtrd 2777 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = 𝐷)
281fvexi 6920 . . . . . . . . . 10 𝑉 ∈ V
29 imaexg 7935 . . . . . . . . . 10 (𝑉 ∈ V → (𝑉 “ ( ∪ {𝑎})) ∈ V)
3028, 29ax-mp 5 . . . . . . . . 9 (𝑉 “ ( ∪ {𝑎})) ∈ V
3130uniex 7761 . . . . . . . 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 769 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = (2nd ‘(1st𝑠)))
359fveq2d 6910 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
36 ot2ndg 8029 . . . . . . . . . . . . . . 15 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3719, 22, 24, 36syl3anc 1373 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3834, 35, 373eqtrd 2781 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = 𝐻)
39 simpr 484 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = (2nd𝑠))
408fveq2d 6910 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
41 ot3rdg 8030 . . . . . . . . . . . . . . . 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 2781 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = 𝐴)
4443sneqd 4638 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → {𝑎} = {𝐴})
4538, 44uneq12d 4169 . . . . . . . . . . . 12 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ( ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
4645imaeq2d 6078 . . . . . . . . . . 11 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
4746unieqd 4920 . . . . . . . . . 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, 48eqtr4di 2795 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = 𝑍)
5033, 49sylan9eqr 2799 . . . . . . . 8 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → 𝑧 = 𝑍)
5150sqxpeqd 5717 . . . . . . 7 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
5232, 51csbied 3935 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑍 × 𝑍))
5327, 52ineq12d 4221 . . . . 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 4888 . . . 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 3935 . . 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 3935 . 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 5470 . . 3 ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
5958a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
605, 56, 57, 59fvmptd 7023 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 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  cun 3949  cin 3950  wss 3951  {csn 4626  cotp 4634   cuni 4907  cmpt 5225   × cxp 5683  ccnv 5684  cima 5688  cfv 6561  1st c1st 8012  2nd c2nd 8013  Fincfn 8985  mExcmex 35472  mDVcmdv 35473  mVarscmvrs 35474  mPreStcmpst 35478  mStRedcmsr 35479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-mpst 35498  df-msr 35499
This theorem is referenced by:  msrf  35547  msrid  35550  elmsta  35553  mthmpps  35587
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