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Theorem elmrsubrn 34114
Description: Characterization of the substitutions as functions from expressions to expressions that distribute under concatenation and map constants to themselves. (The constant part uses (𝐶𝑉) because we don't know that 𝐶 and 𝑉 are disjoint until we get to ismfs 34143.) (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubccat.s 𝑆 = (mRSubst‘𝑇)
mrsubccat.r 𝑅 = (mREx‘𝑇)
mrsubcn.v 𝑉 = (mVR‘𝑇)
mrsubcn.c 𝐶 = (mCN‘𝑇)
Assertion
Ref Expression
elmrsubrn (𝑇𝑊 → (𝐹 ∈ ran 𝑆 ↔ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))))
Distinct variable groups:   𝑥,𝑐,𝑦,𝐶   𝑥,𝑅,𝑦   𝑆,𝑐,𝑥,𝑦   𝑥,𝑇,𝑦   𝐹,𝑐,𝑥,𝑦   𝑉,𝑐,𝑥,𝑦   𝑥,𝑊,𝑦
Allowed substitution hints:   𝑅(𝑐)   𝑇(𝑐)   𝑊(𝑐)

Proof of Theorem elmrsubrn
Dummy variables 𝑟 𝑣 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrsubccat.s . . . 4 𝑆 = (mRSubst‘𝑇)
2 mrsubccat.r . . . 4 𝑅 = (mREx‘𝑇)
31, 2mrsubf 34111 . . 3 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
4 mrsubcn.v . . . . 5 𝑉 = (mVR‘𝑇)
5 mrsubcn.c . . . . 5 𝐶 = (mCN‘𝑇)
61, 2, 4, 5mrsubcn 34113 . . . 4 ((𝐹 ∈ ran 𝑆𝑐 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
76ralrimiva 3143 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
81, 2mrsubccat 34112 . . . . 5 ((𝐹 ∈ ran 𝑆𝑥𝑅𝑦𝑅) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
983expb 1120 . . . 4 ((𝐹 ∈ ran 𝑆 ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
109ralrimivva 3197 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
113, 7, 103jca 1128 . 2 (𝐹 ∈ ran 𝑆 → (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
125, 4, 2mrexval 34095 . . . . . . 7 (𝑇𝑊𝑅 = Word (𝐶𝑉))
1312adantr 481 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑅 = Word (𝐶𝑉))
14 s1eq 14488 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → ⟨“𝑤”⟩ = ⟨“𝑣”⟩)
1514fveq2d 6846 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝐹‘⟨“𝑤”⟩) = (𝐹‘⟨“𝑣”⟩))
16 eqid 2736 . . . . . . . . . . . 12 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))
17 fvex 6855 . . . . . . . . . . . 12 (𝐹‘⟨“𝑣”⟩) ∈ V
1815, 16, 17fvmpt 6948 . . . . . . . . . . 11 (𝑣𝑉 → ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣) = (𝐹‘⟨“𝑣”⟩))
1918adantl 482 . . . . . . . . . 10 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ 𝑣𝑉) → ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣) = (𝐹‘⟨“𝑣”⟩))
20 difun2 4440 . . . . . . . . . . . . . . 15 ((𝐶𝑉) ∖ 𝑉) = (𝐶𝑉)
2120eleq2i 2829 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ 𝑣 ∈ (𝐶𝑉))
22 eldif 3920 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
2321, 22bitr3i 276 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐶𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
24 simpr2 1195 . . . . . . . . . . . . . 14 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
25 s1eq 14488 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑣 → ⟨“𝑐”⟩ = ⟨“𝑣”⟩)
2625fveq2d 6846 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑣 → (𝐹‘⟨“𝑐”⟩) = (𝐹‘⟨“𝑣”⟩))
2726, 25eqeq12d 2752 . . . . . . . . . . . . . . 15 (𝑐 = 𝑣 → ((𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ↔ (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩))
2827rspccva 3580 . . . . . . . . . . . . . 14 ((∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
2924, 28sylan 580 . . . . . . . . . . . . 13 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3023, 29sylan2br 595 . . . . . . . . . . . 12 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3130anassrs 468 . . . . . . . . . . 11 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3231eqcomd 2742 . . . . . . . . . 10 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → ⟨“𝑣”⟩ = (𝐹‘⟨“𝑣”⟩))
3319, 32ifeqda 4522 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩) = (𝐹‘⟨“𝑣”⟩))
3433mpteq2dva 5205 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)))
3534coeq1d 5817 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑟) = ((𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) ∘ 𝑟))
3635oveq2d 7373 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑟)) = ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) ∘ 𝑟)))
3713, 36mpteq12dv 5196 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑟𝑅 ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑟))) = (𝑟 ∈ Word (𝐶𝑉) ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) ∘ 𝑟))))
38 elun2 4137 . . . . . . . 8 (𝑣𝑉𝑣 ∈ (𝐶𝑉))
39 simplr1 1215 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝐹:𝑅𝑅)
40 simpr 485 . . . . . . . . . . 11 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑣 ∈ (𝐶𝑉))
4140s1cld 14491 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ Word (𝐶𝑉))
4212ad2antrr 724 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑅 = Word (𝐶𝑉))
4341, 42eleqtrrd 2841 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ 𝑅)
4439, 43ffvelcdmd 7036 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4538, 44sylan2 593 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4615cbvmptv 5218 . . . . . . 7 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑣𝑉 ↦ (𝐹‘⟨“𝑣”⟩))
4745, 46fmptd 7062 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅)
48 ssid 3966 . . . . . 6 𝑉𝑉
49 eqid 2736 . . . . . . 7 (freeMnd‘(𝐶𝑉)) = (freeMnd‘(𝐶𝑉))
505, 4, 2, 1, 49mrsubfval 34102 . . . . . 6 (((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) = (𝑟𝑅 ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑟))))
5147, 48, 50sylancl 586 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) = (𝑟𝑅 ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑟))))
525fvexi 6856 . . . . . . . . 9 𝐶 ∈ V
534fvexi 6856 . . . . . . . . 9 𝑉 ∈ V
5452, 53unex 7680 . . . . . . . 8 (𝐶𝑉) ∈ V
5549frmdmnd 18669 . . . . . . . 8 ((𝐶𝑉) ∈ V → (freeMnd‘(𝐶𝑉)) ∈ Mnd)
5654, 55ax-mp 5 . . . . . . 7 (freeMnd‘(𝐶𝑉)) ∈ Mnd
5756a1i 11 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (freeMnd‘(𝐶𝑉)) ∈ Mnd)
5854a1i 11 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐶𝑉) ∈ V)
5944, 42eleqtrd 2840 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ Word (𝐶𝑉))
6059fmpttd 7063 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)):(𝐶𝑉)⟶Word (𝐶𝑉))
61 simpr1 1194 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:𝑅𝑅)
6213, 13feq23d 6663 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹:𝑅𝑅𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉)))
6361, 62mpbid 231 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉))
64 simpr3 1196 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
65 simprl 769 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
6612adantr 481 . . . . . . . . . . . . . . . 16 ((𝑇𝑊𝐹:𝑅𝑅) → 𝑅 = Word (𝐶𝑉))
6766adantr 481 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑅 = Word (𝐶𝑉))
6865, 67eleqtrd 2840 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥 ∈ Word (𝐶𝑉))
69 simprr 771 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
7069, 67eleqtrd 2840 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦 ∈ Word (𝐶𝑉))
71 eqid 2736 . . . . . . . . . . . . . . . . . 18 (Base‘(freeMnd‘(𝐶𝑉))) = (Base‘(freeMnd‘(𝐶𝑉)))
7249, 71frmdbas 18662 . . . . . . . . . . . . . . . . 17 ((𝐶𝑉) ∈ V → (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉))
7354, 72ax-mp 5 . . . . . . . . . . . . . . . 16 (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉)
7473eqcomi 2745 . . . . . . . . . . . . . . 15 Word (𝐶𝑉) = (Base‘(freeMnd‘(𝐶𝑉)))
75 eqid 2736 . . . . . . . . . . . . . . 15 (+g‘(freeMnd‘(𝐶𝑉))) = (+g‘(freeMnd‘(𝐶𝑉)))
7649, 74, 75frmdadd 18665 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (𝐶𝑉) ∧ 𝑦 ∈ Word (𝐶𝑉)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
7768, 70, 76syl2anc 584 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
7877fveq2d 6846 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = (𝐹‘(𝑥 ++ 𝑦)))
79 ffvelcdm 7032 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑥𝑅) → (𝐹𝑥) ∈ 𝑅)
8079ad2ant2lr 746 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ 𝑅)
8180, 67eleqtrd 2840 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ Word (𝐶𝑉))
82 ffvelcdm 7032 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑦𝑅) → (𝐹𝑦) ∈ 𝑅)
8382ad2ant2l 744 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ 𝑅)
8483, 67eleqtrd 2840 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ Word (𝐶𝑉))
8549, 74, 75frmdadd 18665 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ Word (𝐶𝑉) ∧ (𝐹𝑦) ∈ Word (𝐶𝑉)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8681, 84, 85syl2anc 584 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8778, 86eqeq12d 2752 . . . . . . . . . . 11 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
88872ralbidva 3210 . . . . . . . . . 10 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
8966raleqdv 3313 . . . . . . . . . . 11 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦))))
9066, 89raleqbidv 3319 . . . . . . . . . 10 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑥 ∈ Word (𝐶𝑉)∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦))))
9188, 90bitr3d 280 . . . . . . . . 9 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ ∀𝑥 ∈ Word (𝐶𝑉)∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦))))
92913ad2antr1 1188 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ ∀𝑥 ∈ Word (𝐶𝑉)∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦))))
9364, 92mpbid 231 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑥 ∈ Word (𝐶𝑉)∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)))
94 wrd0 14427 . . . . . . . . . . . 12 ∅ ∈ Word (𝐶𝑉)
95 ffvelcdm 7032 . . . . . . . . . . . 12 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ ∅ ∈ Word (𝐶𝑉)) → (𝐹‘∅) ∈ Word (𝐶𝑉))
9663, 94, 95sylancl 586 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) ∈ Word (𝐶𝑉))
97 lencl 14421 . . . . . . . . . . 11 ((𝐹‘∅) ∈ Word (𝐶𝑉) → (♯‘(𝐹‘∅)) ∈ ℕ0)
9896, 97syl 17 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) ∈ ℕ0)
9998nn0cnd 12475 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) ∈ ℂ)
100 0cnd 11148 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 0 ∈ ℂ)
10199addid1d 11355 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((♯‘(𝐹‘∅)) + 0) = (♯‘(𝐹‘∅)))
10294, 13eleqtrrid 2845 . . . . . . . . . . . 12 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∅ ∈ 𝑅)
103 fvoveq1 7380 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (𝐹‘(𝑥 ++ 𝑦)) = (𝐹‘(∅ ++ 𝑦)))
104 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
105104oveq1d 7372 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝐹𝑥) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)))
106103, 105eqeq12d 2752 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ (𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦))))
107 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (∅ ++ 𝑦) = (∅ ++ ∅))
108 ccatidid 14478 . . . . . . . . . . . . . . . 16 (∅ ++ ∅) = ∅
109107, 108eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∅ ++ 𝑦) = ∅)
110109fveq2d 6846 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐹‘(∅ ++ 𝑦)) = (𝐹‘∅))
111 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
112111oveq2d 7373 . . . . . . . . . . . . . 14 (𝑦 = ∅ → ((𝐹‘∅) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹‘∅)))
113110, 112eqeq12d 2752 . . . . . . . . . . . . 13 (𝑦 = ∅ → ((𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)) ↔ (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅))))
114106, 113rspc2va 3591 . . . . . . . . . . . 12 (((∅ ∈ 𝑅 ∧ ∅ ∈ 𝑅) ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
115102, 102, 64, 114syl21anc 836 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
116115fveq2d 6846 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) = (♯‘((𝐹‘∅) ++ (𝐹‘∅))))
117 ccatlen 14463 . . . . . . . . . . 11 (((𝐹‘∅) ∈ Word (𝐶𝑉) ∧ (𝐹‘∅) ∈ Word (𝐶𝑉)) → (♯‘((𝐹‘∅) ++ (𝐹‘∅))) = ((♯‘(𝐹‘∅)) + (♯‘(𝐹‘∅))))
11896, 96, 117syl2anc 584 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘((𝐹‘∅) ++ (𝐹‘∅))) = ((♯‘(𝐹‘∅)) + (♯‘(𝐹‘∅))))
119101, 116, 1183eqtrrd 2781 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((♯‘(𝐹‘∅)) + (♯‘(𝐹‘∅))) = ((♯‘(𝐹‘∅)) + 0))
12099, 99, 100, 119addcanad 11360 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) = 0)
121 fvex 6855 . . . . . . . . 9 (𝐹‘∅) ∈ V
122 hasheq0 14263 . . . . . . . . 9 ((𝐹‘∅) ∈ V → ((♯‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅))
123121, 122ax-mp 5 . . . . . . . 8 ((♯‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅)
124120, 123sylib 217 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ∅)
12556, 56pm3.2i 471 . . . . . . . 8 ((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (freeMnd‘(𝐶𝑉)) ∈ Mnd)
12649frmd0 18670 . . . . . . . . 9 ∅ = (0g‘(freeMnd‘(𝐶𝑉)))
12774, 74, 75, 75, 126, 126ismhm 18603 . . . . . . . 8 (𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))) ↔ (((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (freeMnd‘(𝐶𝑉)) ∈ Mnd) ∧ (𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ ∀𝑥 ∈ Word (𝐶𝑉)∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ∧ (𝐹‘∅) = ∅)))
128125, 127mpbiran 707 . . . . . . 7 (𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))) ↔ (𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ ∀𝑥 ∈ Word (𝐶𝑉)∀𝑦 ∈ Word (𝐶𝑉)(𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ∧ (𝐹‘∅) = ∅))
12963, 93, 124, 128syl3anbrc 1343 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))))
130 eqid 2736 . . . . . . . . . 10 (varFMnd‘(𝐶𝑉)) = (varFMnd‘(𝐶𝑉))
131130vrmdf 18668 . . . . . . . . 9 ((𝐶𝑉) ∈ V → (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉))
13254, 131ax-mp 5 . . . . . . . 8 (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)
133 fcompt 7079 . . . . . . . 8 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))))
13463, 132, 133sylancl 586 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))))
135130vrmdval 18667 . . . . . . . . . 10 (((𝐶𝑉) ∈ V ∧ 𝑣 ∈ (𝐶𝑉)) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
13654, 135mpan 688 . . . . . . . . 9 (𝑣 ∈ (𝐶𝑉) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
137136fveq2d 6846 . . . . . . . 8 (𝑣 ∈ (𝐶𝑉) → (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣)) = (𝐹‘⟨“𝑣”⟩))
138137mpteq2ia 5208 . . . . . . 7 (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩))
139134, 138eqtrdi 2792 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)))
14049, 74, 130frmdup3lem 18676 . . . . . 6 ((((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (𝐶𝑉) ∈ V ∧ (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)):(𝐶𝑉)⟶Word (𝐶𝑉)) ∧ (𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))) ∧ (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)))) → 𝐹 = (𝑟 ∈ Word (𝐶𝑉) ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) ∘ 𝑟))))
14157, 58, 60, 129, 139, 140syl32anc 1378 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 = (𝑟 ∈ Word (𝐶𝑉) ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) ∘ 𝑟))))
14237, 51, 1413eqtr4rd 2787 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 = (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))))
1434, 2, 1mrsubff 34106 . . . . . . 7 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
144143ffnd 6669 . . . . . 6 (𝑇𝑊𝑆 Fn (𝑅pm 𝑉))
145144adantr 481 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑆 Fn (𝑅pm 𝑉))
1462fvexi 6856 . . . . . . 7 𝑅 ∈ V
147 elpm2r 8783 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉)) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
148146, 53, 147mpanl12 700 . . . . . 6 (((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
14947, 48, 148sylancl 586 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
150 fnfvelrn 7031 . . . . 5 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
151145, 149, 150syl2anc 584 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
152142, 151eqeltrd 2838 . . 3 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ran 𝑆)
153152ex 413 . 2 (𝑇𝑊 → ((𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → 𝐹 ∈ ran 𝑆))
15411, 153impbid2 225 1 (𝑇𝑊 → (𝐹 ∈ ran 𝑆 ↔ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  Vcvv 3445  cdif 3907  cun 3908  wss 3910  c0 4282  ifcif 4486  cmpt 5188  ran crn 5634  ccom 5637   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  pm cpm 8766  0cc0 11051   + caddc 11054  0cn0 12413  chash 14230  Word cword 14402   ++ cconcat 14458  ⟨“cs1 14483  Basecbs 17083  +gcplusg 17133   Σg cgsu 17322  Mndcmnd 18556   MndHom cmhm 18599  freeMndcfrmd 18657  varFMndcvrmd 18658  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:  mrsubco  34115
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