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Theorem mrsubco 34115
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 2736 . . . . 5 (mREx‘𝑇) = (mREx‘𝑇)
31, 2mrsubf 34111 . . . 4 (𝐹 ∈ ran 𝑆𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
43adantr 481 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
51, 2mrsubf 34111 . . . 4 (𝐺 ∈ ran 𝑆𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
65adantl 482 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7 fco 6692 . . 3 ((𝐹:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇)) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
84, 6, 7syl2anc 584 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
96adantr 481 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
10 eldifi 4086 . . . . . . . . 9 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11 elun1 4136 . . . . . . . . 9 (𝑐 ∈ (mCN‘𝑇) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1210, 11syl 17 . . . . . . . 8 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1312adantl 482 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1413s1cld 14491 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15 n0i 4293 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
161rnfvprc 6836 . . . . . . . . . 10 𝑇 ∈ V → ran 𝑆 = ∅)
1715, 16nsyl2 141 . . . . . . . . 9 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
1817adantr 481 . . . . . . . 8 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝑇 ∈ V)
1918adantr 481 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑇 ∈ V)
20 eqid 2736 . . . . . . . 8 (mCN‘𝑇) = (mCN‘𝑇)
21 eqid 2736 . . . . . . . 8 (mVR‘𝑇) = (mVR‘𝑇)
2220, 21, 2mrexval 34095 . . . . . . 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 2841 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
25 fvco3 6940 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ⟨“𝑐”⟩ ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
269, 24, 25syl2anc 584 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
271, 2, 21, 20mrsubcn 34113 . . . . . 6 ((𝐺 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
2827adantll 712 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
2928fveq2d 6846 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
301, 2, 21, 20mrsubcn 34113 . . . . 5 ((𝐹 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3130adantlr 713 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3226, 29, 313eqtrd 2780 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3332ralrimiva 3143 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
341, 2mrsubccat 34112 . . . . . . . 8 ((𝐺 ∈ ran 𝑆𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
35343expb 1120 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
3635adantll 712 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
3736fveq2d 6846 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))))
38 simpll 765 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐹 ∈ ran 𝑆)
396adantr 481 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
40 simprl 769 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ (mREx‘𝑇))
4139, 40ffvelcdmd 7036 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑥) ∈ (mREx‘𝑇))
42 simprr 771 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ (mREx‘𝑇))
4339, 42ffvelcdmd 7036 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑦) ∈ (mREx‘𝑇))
441, 2mrsubccat 34112 . . . . . 6 ((𝐹 ∈ ran 𝑆 ∧ (𝐺𝑥) ∈ (mREx‘𝑇) ∧ (𝐺𝑦) ∈ (mREx‘𝑇)) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4538, 41, 43, 44syl3anc 1371 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4637, 45eqtrd 2776 . . . 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 481 . . . . . . . 8 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
4940, 48eleqtrd 2840 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5042, 48eleqtrd 2840 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
51 ccatcl 14462 . . . . . . 7 ((𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5249, 50, 51syl2anc 584 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5352, 48eleqtrrd 2841 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
54 fvco3 6940 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (𝑥 ++ 𝑦) ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
5539, 53, 54syl2anc 584 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
56 fvco3 6940 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑥 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
5739, 40, 56syl2anc 584 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
58 fvco3 6940 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
5939, 42, 58syl2anc 584 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
6057, 59oveq12d 7375 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
6146, 55, 603eqtr4d 2786 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
6261ralrimivva 3197 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
631, 2, 21, 20elmrsubrn 34114 . . 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 1342 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  cun 3908  c0 4282  ran crn 5634  ccom 5637  wf 6492  cfv 6496  (class class class)co 7357  Word cword 14402   ++ cconcat 14458  ⟨“cs1 14483  mCNcmcn 34054  mVRcmvar 34055  mRExcmrex 34060  mRSubstcmrsub 34064
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-lsw 14451  df-concat 14459  df-s1 14484  df-substr 14529  df-pfx 14559  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-frmd 18659  df-vrmd 18660  df-mrex 34080  df-mrsub 34084
This theorem is referenced by:  msubco  34125
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