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Theorem mrsubco 32876
Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mrsubco.s 𝑆 = (mRSubst‘𝑇)
Assertion
Ref Expression
mrsubco ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)

Proof of Theorem mrsubco
Dummy variables 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrsubco.s . . . . 5 𝑆 = (mRSubst‘𝑇)
2 eqid 2801 . . . . 5 (mREx‘𝑇) = (mREx‘𝑇)
31, 2mrsubf 32872 . . . 4 (𝐹 ∈ ran 𝑆𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
43adantr 484 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
51, 2mrsubf 32872 . . . 4 (𝐺 ∈ ran 𝑆𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
65adantl 485 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7 fco 6509 . . 3 ((𝐹:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇)) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
84, 6, 7syl2anc 587 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
96adantr 484 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
10 eldifi 4057 . . . . . . . . 9 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11 elun1 4106 . . . . . . . . 9 (𝑐 ∈ (mCN‘𝑇) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1210, 11syl 17 . . . . . . . 8 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1312adantl 485 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1413s1cld 13952 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15 n0i 4252 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
161rnfvprc 6643 . . . . . . . . . 10 𝑇 ∈ V → ran 𝑆 = ∅)
1715, 16nsyl2 143 . . . . . . . . 9 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
1817adantr 484 . . . . . . . 8 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝑇 ∈ V)
1918adantr 484 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑇 ∈ V)
20 eqid 2801 . . . . . . . 8 (mCN‘𝑇) = (mCN‘𝑇)
21 eqid 2801 . . . . . . . 8 (mVR‘𝑇) = (mVR‘𝑇)
2220, 21, 2mrexval 32856 . . . . . . 7 (𝑇 ∈ V → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2319, 22syl 17 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2414, 23eleqtrrd 2896 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
25 fvco3 6741 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ⟨“𝑐”⟩ ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
269, 24, 25syl2anc 587 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
271, 2, 21, 20mrsubcn 32874 . . . . . 6 ((𝐺 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
2827adantll 713 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
2928fveq2d 6653 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
301, 2, 21, 20mrsubcn 32874 . . . . 5 ((𝐹 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3130adantlr 714 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3226, 29, 313eqtrd 2840 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3332ralrimiva 3152 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
341, 2mrsubccat 32873 . . . . . . . 8 ((𝐺 ∈ ran 𝑆𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
35343expb 1117 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
3635adantll 713 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
3736fveq2d 6653 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))))
38 simpll 766 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐹 ∈ ran 𝑆)
396adantr 484 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
40 simprl 770 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ (mREx‘𝑇))
4139, 40ffvelrnd 6833 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑥) ∈ (mREx‘𝑇))
42 simprr 772 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ (mREx‘𝑇))
4339, 42ffvelrnd 6833 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑦) ∈ (mREx‘𝑇))
441, 2mrsubccat 32873 . . . . . 6 ((𝐹 ∈ ran 𝑆 ∧ (𝐺𝑥) ∈ (mREx‘𝑇) ∧ (𝐺𝑦) ∈ (mREx‘𝑇)) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4538, 41, 43, 44syl3anc 1368 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4637, 45eqtrd 2836 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4718, 22syl 17 . . . . . . . . 9 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
4847adantr 484 . . . . . . . 8 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
4940, 48eleqtrd 2895 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5042, 48eleqtrd 2895 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
51 ccatcl 13921 . . . . . . 7 ((𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5249, 50, 51syl2anc 587 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5352, 48eleqtrrd 2896 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
54 fvco3 6741 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (𝑥 ++ 𝑦) ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
5539, 53, 54syl2anc 587 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
56 fvco3 6741 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑥 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
5739, 40, 56syl2anc 587 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
58 fvco3 6741 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
5939, 42, 58syl2anc 587 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
6057, 59oveq12d 7157 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
6146, 55, 603eqtr4d 2846 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
6261ralrimivva 3159 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
631, 2, 21, 20elmrsubrn 32875 . . 3 (𝑇 ∈ V → ((𝐹𝐺) ∈ ran 𝑆 ↔ ((𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))))
6418, 63syl 17 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ((𝐹𝐺) ∈ ran 𝑆 ↔ ((𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))))
658, 33, 62, 64mpbir3and 1339 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109  Vcvv 3444  cdif 3881  cun 3882  c0 4246  ran crn 5524  ccom 5527  wf 6324  cfv 6328  (class class class)co 7139  Word cword 13861   ++ cconcat 13917  ⟨“cs1 13944  mCNcmcn 32815  mVRcmvar 32816  mRExcmrex 32821  mRSubstcmrsub 32825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-pm 8396  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-seq 13369  df-hash 13691  df-word 13862  df-lsw 13910  df-concat 13918  df-s1 13945  df-substr 13998  df-pfx 14028  df-struct 16480  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-0g 16710  df-gsum 16711  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-mhm 17951  df-submnd 17952  df-frmd 18009  df-vrmd 18010  df-mrex 32841  df-mrsub 32845
This theorem is referenced by:  msubco  32886
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