Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mrsubco Structured version   Visualization version   GIF version

Theorem mrsubco 31756
 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 2771 . . . . 5 (mREx‘𝑇) = (mREx‘𝑇)
31, 2mrsubf 31752 . . . 4 (𝐹 ∈ ran 𝑆𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
43adantr 466 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐹:(mREx‘𝑇)⟶(mREx‘𝑇))
51, 2mrsubf 31752 . . . 4 (𝐺 ∈ ran 𝑆𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
65adantl 467 . . 3 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
7 fco 6198 . . 3 ((𝐹:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇)) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
84, 6, 7syl2anc 573 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺):(mREx‘𝑇)⟶(mREx‘𝑇))
96adantr 466 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
10 eldifi 3883 . . . . . . . . 9 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ (mCN‘𝑇))
11 elun1 3931 . . . . . . . . 9 (𝑐 ∈ (mCN‘𝑇) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1210, 11syl 17 . . . . . . . 8 (𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇)) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1312adantl 467 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑐 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
1413s1cld 13583 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
15 n0i 4068 . . . . . . . . . 10 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
16 fvprc 6326 . . . . . . . . . . . . 13 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
171, 16syl5eq 2817 . . . . . . . . . . . 12 𝑇 ∈ V → 𝑆 = ∅)
1817rneqd 5491 . . . . . . . . . . 11 𝑇 ∈ V → ran 𝑆 = ran ∅)
19 rn0 5515 . . . . . . . . . . 11 ran ∅ = ∅
2018, 19syl6eq 2821 . . . . . . . . . 10 𝑇 ∈ V → ran 𝑆 = ∅)
2115, 20nsyl2 144 . . . . . . . . 9 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
2221adantr 466 . . . . . . . 8 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → 𝑇 ∈ V)
2322adantr 466 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → 𝑇 ∈ V)
24 eqid 2771 . . . . . . . 8 (mCN‘𝑇) = (mCN‘𝑇)
25 eqid 2771 . . . . . . . 8 (mVR‘𝑇) = (mVR‘𝑇)
2624, 25, 2mrexval 31736 . . . . . . 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 2853 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ⟨“𝑐”⟩ ∈ (mREx‘𝑇))
29 fvco3 6417 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ ⟨“𝑐”⟩ ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
309, 28, 29syl2anc 573 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = (𝐹‘(𝐺‘⟨“𝑐”⟩)))
311, 2, 25, 24mrsubcn 31754 . . . . . 6 ((𝐺 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3231adantll 693 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐺‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3332fveq2d 6336 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘(𝐺‘⟨“𝑐”⟩)) = (𝐹‘⟨“𝑐”⟩))
341, 2, 25, 24mrsubcn 31754 . . . . 5 ((𝐹 ∈ ran 𝑆𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3534adantlr 694 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3630, 33, 353eqtrd 2809 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ 𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))) → ((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
3736ralrimiva 3115 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑐 ∈ ((mCN‘𝑇) ∖ (mVR‘𝑇))((𝐹𝐺)‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
381, 2mrsubccat 31753 . . . . . . . 8 ((𝐺 ∈ ran 𝑆𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
39383expb 1113 . . . . . . 7 ((𝐺 ∈ ran 𝑆 ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
4039adantll 693 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺‘(𝑥 ++ 𝑦)) = ((𝐺𝑥) ++ (𝐺𝑦)))
4140fveq2d 6336 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘(𝐺‘(𝑥 ++ 𝑦))) = (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))))
42 simpll 750 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐹 ∈ ran 𝑆)
436adantr 466 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝐺:(mREx‘𝑇)⟶(mREx‘𝑇))
44 simprl 754 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ (mREx‘𝑇))
4543, 44ffvelrnd 6503 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑥) ∈ (mREx‘𝑇))
46 simprr 756 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ (mREx‘𝑇))
4743, 46ffvelrnd 6503 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐺𝑦) ∈ (mREx‘𝑇))
481, 2mrsubccat 31753 . . . . . 6 ((𝐹 ∈ ran 𝑆 ∧ (𝐺𝑥) ∈ (mREx‘𝑇) ∧ (𝐺𝑦) ∈ (mREx‘𝑇)) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
4942, 45, 47, 48syl3anc 1476 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝐹‘((𝐺𝑥) ++ (𝐺𝑦))) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
5041, 49eqtrd 2805 . . . 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 466 . . . . . . . 8 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (mREx‘𝑇) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5344, 52eleqtrd 2852 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5446, 52eleqtrd 2852 . . . . . . 7 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
55 ccatcl 13556 . . . . . . 7 ((𝑥 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑦 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5653, 54, 55syl2anc 573 . . . . . 6 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5756, 52eleqtrrd 2853 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (𝑥 ++ 𝑦) ∈ (mREx‘𝑇))
58 fvco3 6417 . . . . 5 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ (𝑥 ++ 𝑦) ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
5943, 57, 58syl2anc 573 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (𝐹‘(𝐺‘(𝑥 ++ 𝑦))))
60 fvco3 6417 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑥 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
6143, 44, 60syl2anc 573 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑥) = (𝐹‘(𝐺𝑥)))
62 fvco3 6417 . . . . . 6 ((𝐺:(mREx‘𝑇)⟶(mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇)) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
6343, 46, 62syl2anc 573 . . . . 5 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘𝑦) = (𝐹‘(𝐺𝑦)))
6461, 63oveq12d 6811 . . . 4 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)) = ((𝐹‘(𝐺𝑥)) ++ (𝐹‘(𝐺𝑦))))
6550, 59, 643eqtr4d 2815 . . 3 (((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) ∧ (𝑥 ∈ (mREx‘𝑇) ∧ 𝑦 ∈ (mREx‘𝑇))) → ((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
6665ralrimivva 3120 . 2 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → ∀𝑥 ∈ (mREx‘𝑇)∀𝑦 ∈ (mREx‘𝑇)((𝐹𝐺)‘(𝑥 ++ 𝑦)) = (((𝐹𝐺)‘𝑥) ++ ((𝐹𝐺)‘𝑦)))
671, 2, 25, 24elmrsubrn 31755 . . 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 1427 1 ((𝐹 ∈ ran 𝑆𝐺 ∈ ran 𝑆) → (𝐹𝐺) ∈ ran 𝑆)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∀wral 3061  Vcvv 3351   ∖ cdif 3720   ∪ cun 3721  ∅c0 4063  ran crn 5250   ∘ ccom 5253  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793  Word cword 13487   ++ cconcat 13489  ⟨“cs1 13490  mCNcmcn 31695  mVRcmvar 31696  mRExcmrex 31701  mRSubstcmrsub 31705 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-2 11281  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-word 13495  df-lsw 13496  df-concat 13497  df-s1 13498  df-substr 13499  df-struct 16066  df-ndx 16067  df-slot 16068  df-base 16070  df-sets 16071  df-ress 16072  df-plusg 16162  df-0g 16310  df-gsum 16311  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mhm 17543  df-submnd 17544  df-frmd 17594  df-vrmd 17595  df-mrex 31721  df-mrsub 31725 This theorem is referenced by:  msubco  31766
 Copyright terms: Public domain W3C validator