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Theorem msrval 33009
 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 33008 . . 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 6674 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) → (2nd ‘(1st𝑠)) ∈ V)
7 fvexd 6674 . . . 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 6663 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st𝑠) = (1st ‘⟨𝐷, 𝐻, 𝐴⟩))
109fveq2d 6663 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
11 eqid 2759 . . . . . . . . . . . . 13 (mDV‘𝑇) = (mDV‘𝑇)
12 eqid 2759 . . . . . . . . . . . . 13 (mEx‘𝑇) = (mEx‘𝑇)
1311, 12, 2elmpst 33007 . . . . . . . . . . . 12 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷) ∧ (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mEx‘𝑇)))
1413simp1bi 1143 . . . . . . . . . . 11 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ⊆ (mDV‘𝑇) ∧ 𝐷 = 𝐷))
1514simpld 499 . . . . . . . . . 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 6672 . . . . . . . . . 10 (mDV‘𝑇) ∈ V
1817ssex 5192 . . . . . . . . 9 (𝐷 ⊆ (mDV‘𝑇) → 𝐷 ∈ V)
1916, 18syl 17 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐷 ∈ V)
2013simp2bi 1144 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐻 ⊆ (mEx‘𝑇) ∧ 𝐻 ∈ Fin))
2120simprd 500 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐻 ∈ Fin)
2221ad3antrrr 730 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝐻 ∈ Fin)
2313simp3bi 1145 . . . . . . . . 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 7708 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2619, 22, 24, 25syl3anc 1369 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐷)
2710, 26eqtrd 2794 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (1st ‘(1st𝑠)) = 𝐷)
281fvexi 6673 . . . . . . . . . 10 𝑉 ∈ V
29 imaexg 7626 . . . . . . . . . 10 (𝑉 ∈ V → (𝑉 “ ( ∪ {𝑎})) ∈ V)
3028, 29ax-mp 5 . . . . . . . . 9 (𝑉 “ ( ∪ {𝑎})) ∈ V
3130uniex 7466 . . . . . . . 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 6663 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st𝑠)) = (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)))
36 ot2ndg 7709 . . . . . . . . . . . . . . 15 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mEx‘𝑇)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3719, 22, 24, 36syl3anc 1369 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd ‘(1st ‘⟨𝐷, 𝐻, 𝐴⟩)) = 𝐻)
3834, 35, 373eqtrd 2798 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → = 𝐻)
39 simpr 489 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = (2nd𝑠))
408fveq2d 6663 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (2nd𝑠) = (2nd ‘⟨𝐷, 𝐻, 𝐴⟩))
41 ot3rdg 7710 . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → 𝑎 = 𝐴)
4443sneqd 4535 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → {𝑎} = {𝐴})
4538, 44uneq12d 4070 . . . . . . . . . . . 12 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → ( ∪ {𝑎}) = (𝐻 ∪ {𝐴}))
4645imaeq2d 5902 . . . . . . . . . . 11 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = (𝑉 “ (𝐻 ∪ {𝐴})))
4746unieqd 4813 . . . . . . . . . 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 2812 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) = 𝑍)
5033, 49sylan9eqr 2816 . . . . . . . 8 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → 𝑧 = 𝑍)
5150sqxpeqd 5557 . . . . . . 7 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) ∧ 𝑧 = (𝑉 “ ( ∪ {𝑎}))) → (𝑧 × 𝑧) = (𝑍 × 𝑍))
5232, 51csbied 3842 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ = (2nd ‘(1st𝑠))) ∧ 𝑎 = (2nd𝑠)) → (𝑉 “ ( ∪ {𝑎})) / 𝑧(𝑧 × 𝑧) = (𝑍 × 𝑍))
5327, 52ineq12d 4119 . . . . 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 4779 . . . 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 3842 . . 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 3842 . 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 5326 . . 3 ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V
5958a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨(𝐷 ∩ (𝑍 × 𝑍)), 𝐻, 𝐴⟩ ∈ V)
605, 56, 57, 59fvmptd 6767 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 400   = wceq 1539   ∈ wcel 2112  Vcvv 3410  ⦋csb 3806   ∪ cun 3857   ∩ cin 3858   ⊆ wss 3859  {csn 4523  ⟨cotp 4531  ∪ cuni 4799   ↦ cmpt 5113   × cxp 5523  ◡ccnv 5524   “ cima 5528  ‘cfv 6336  1st c1st 7692  2nd c2nd 7693  Fincfn 8528  mExcmex 32938  mDVcmdv 32939  mVarscmvrs 32940  mPreStcmpst 32944  mStRedcmsr 32945 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-op 4530  df-ot 4532  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5431  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-1st 7694  df-2nd 7695  df-mpst 32964  df-msr 32965 This theorem is referenced by:  msrf  33013  msrid  33016  elmsta  33019  mthmpps  33053
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