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Theorem mrsubco 32665
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 2818 . . . . 5 (mREx‘𝑇) = (mREx‘𝑇)
31, 2mrsubf 32661 . . . 4 (𝐹 ∈ ran 𝑆𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
43adantr 481 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
51, 2mrsubf 32661 . . . 4 (𝐺 ∈ ran 𝑆𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
65adantl 482 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7 fco 6524 . . 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 4100 . . . . . . . . 9 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11 elun1 4149 . . . . . . . . 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 13945 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15 n0i 4296 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
161rnfvprc 6657 . . . . . . . . . 10 𝑇 ∈ V → ran 𝑆 = ∅)
1715, 16nsyl2 143 . . . . . . . . 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 2818 . . . . . . . 8 (mCN‘𝑇) = (mCN‘𝑇)
21 eqid 2818 . . . . . . . 8 (mVR‘𝑇) = (mVR‘𝑇)
2220, 21, 2mrexval 32645 . . . . . . 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 2913 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
25 fvco3 6753 . . . . 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 32663 . . . . . 6 ((𝐺 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
2827adantll 710 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
2928fveq2d 6667 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
301, 2, 21, 20mrsubcn 32663 . . . . 5 ((𝐹 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3130adantlr 711 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3226, 29, 313eqtrd 2857 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3332ralrimiva 3179 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
341, 2mrsubccat 32662 . . . . . . . 8 ((𝐺 ∈ ran 𝑆𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
35343expb 1112 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
3635adantll 710 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
3736fveq2d 6667 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))))
38 simpll 763 . . . . . 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 767 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ (mREx‘𝑇))
4139, 40ffvelrnd 6844 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑥) ∈ (mREx‘𝑇))
42 simprr 769 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ (mREx‘𝑇))
4339, 42ffvelrnd 6844 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑦) ∈ (mREx‘𝑇))
441, 2mrsubccat 32662 . . . . . 6 ((𝐹 ∈ ran 𝑆 ∧ (𝐺𝑥) ∈ (mREx‘𝑇) ∧ (𝐺𝑦) ∈ (mREx‘𝑇)) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4538, 41, 43, 44syl3anc 1363 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4637, 45eqtrd 2853 . . . 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 2912 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5042, 48eleqtrd 2912 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
51 ccatcl 13914 . . . . . . 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 2913 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
54 fvco3 6753 . . . . 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 6753 . . . . . 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 6753 . . . . . 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 7163 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
6146, 55, 603eqtr4d 2863 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
6261ralrimivva 3188 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
631, 2, 21, 20elmrsubrn 32664 . . 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 1334 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  cdif 3930  cun 3931  c0 4288  ran crn 5549  ccom 5552  wf 6344  cfv 6348  (class class class)co 7145  Word cword 13849   ++ cconcat 13910  ⟨“cs1 13937  mCNcmcn 32604  mVRcmvar 32605  mRExcmrex 32610  mRSubstcmrsub 32614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mhm 17944  df-submnd 17945  df-frmd 18002  df-vrmd 18003  df-mrex 32630  df-mrsub 32634
This theorem is referenced by:  msubco  32675
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