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Theorem mrsubco 31179
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 2621 . . . . 5 (mREx‘𝑇) = (mREx‘𝑇)
31, 2mrsubf 31175 . . . 4 (𝐹 ∈ ran 𝑆𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
43adantr 481 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
51, 2mrsubf 31175 . . . 4 (𝐺 ∈ ran 𝑆𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
65adantl 482 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7 fco 6025 . . 3 ((𝐹:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇)) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
84, 6, 7syl2anc 692 . 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 3716 . . . . . . . . 9 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11 elun1 3764 . . . . . . . . 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 13338 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15 n0i 3902 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
16 fvprc 6152 . . . . . . . . . . . . 13 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
171, 16syl5eq 2667 . . . . . . . . . . . 12 𝑇 ∈ V → 𝑆 = ∅)
1817rneqd 5323 . . . . . . . . . . 11 𝑇 ∈ V → ran 𝑆 = ran ∅)
19 rn0 5347 . . . . . . . . . . 11 ran ∅ = ∅
2018, 19syl6eq 2671 . . . . . . . . . 10 𝑇 ∈ V → ran 𝑆 = ∅)
2115, 20nsyl2 142 . . . . . . . . 9 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
2221adantr 481 . . . . . . . 8 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝑇 ∈ V)
2322adantr 481 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑇 ∈ V)
24 eqid 2621 . . . . . . . 8 (mCN‘𝑇) = (mCN‘𝑇)
25 eqid 2621 . . . . . . . 8 (mVR‘𝑇) = (mVR‘𝑇)
2624, 25, 2mrexval 31159 . . . . . . 7 (𝑇 ∈ V → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2723, 26syl 17 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2814, 27eleqtrrd 2701 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
29 fvco3 6242 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ⟨“𝑐”⟩ ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
309, 28, 29syl2anc 692 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
311, 2, 25, 24mrsubcn 31177 . . . . . 6 ((𝐺 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3231adantll 749 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3332fveq2d 6162 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
341, 2, 25, 24mrsubcn 31177 . . . . 5 ((𝐹 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3534adantlr 750 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3630, 33, 353eqtrd 2659 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3736ralrimiva 2962 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
381, 2mrsubccat 31176 . . . . . . . 8 ((𝐺 ∈ ran 𝑆𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
39383expb 1263 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
4039adantll 749 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
4140fveq2d 6162 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))))
42 simpll 789 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐹 ∈ ran 𝑆)
436adantr 481 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
44 simprl 793 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ (mREx‘𝑇))
4543, 44ffvelrnd 6326 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑥) ∈ (mREx‘𝑇))
46 simprr 795 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ (mREx‘𝑇))
4743, 46ffvelrnd 6326 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑦) ∈ (mREx‘𝑇))
481, 2mrsubccat 31176 . . . . . 6 ((𝐹 ∈ ran 𝑆 ∧ (𝐺𝑥) ∈ (mREx‘𝑇) ∧ (𝐺𝑦) ∈ (mREx‘𝑇)) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4942, 45, 47, 48syl3anc 1323 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
5041, 49eqtrd 2655 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
5122, 26syl 17 . . . . . . . . 9 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5251adantr 481 . . . . . . . 8 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5344, 52eleqtrd 2700 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5446, 52eleqtrd 2700 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
55 ccatcl 13314 . . . . . . 7 ((𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5653, 54, 55syl2anc 692 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5756, 52eleqtrrd 2701 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
58 fvco3 6242 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (𝑥 ++ 𝑦) ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
5943, 57, 58syl2anc 692 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
60 fvco3 6242 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑥 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
6143, 44, 60syl2anc 692 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
62 fvco3 6242 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
6343, 46, 62syl2anc 692 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
6461, 63oveq12d 6633 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
6550, 59, 643eqtr4d 2665 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
6665ralrimivva 2967 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
671, 2, 25, 24elmrsubrn 31178 . . 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)))))
6822, 67syl 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)))))
698, 37, 66, 68mpbir3and 1243 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cdif 3557  cun 3558  c0 3897  ran crn 5085  ccom 5088  wf 5853  cfv 5857  (class class class)co 6615  Word cword 13246   ++ cconcat 13248  ⟨“cs1 13249  mCNcmcn 31118  mVRcmvar 31119  mRExcmrex 31124  mRSubstcmrsub 31128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-n0 11253  df-xnn0 11324  df-z 11338  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-word 13254  df-lsw 13255  df-concat 13256  df-s1 13257  df-substr 13258  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-0g 16042  df-gsum 16043  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276  df-frmd 17326  df-vrmd 17327  df-mrex 31144  df-mrsub 31148
This theorem is referenced by:  msubco  31189
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