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Theorem elmrsubrn 31160
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 31189.) (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 31157 . . 3 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
4 mrsubcn.v . . . . 5 𝑉 = (mVR‘𝑇)
5 mrsubcn.c . . . . 5 𝐶 = (mCN‘𝑇)
61, 2, 4, 5mrsubcn 31159 . . . 4 ((𝐹 ∈ ran 𝑆𝑐 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
76ralrimiva 2961 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
81, 2mrsubccat 31158 . . . . 5 ((𝐹 ∈ ran 𝑆𝑥𝑅𝑦𝑅) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
983expb 1263 . . . 4 ((𝐹 ∈ ran 𝑆 ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
109ralrimivva 2966 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
113, 7, 103jca 1240 . 2 (𝐹 ∈ ran 𝑆 → (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
125, 4, 2mrexval 31141 . . . . . . 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 13327 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → ⟨“𝑤”⟩ = ⟨“𝑣”⟩)
1514fveq2d 6157 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝐹‘⟨“𝑤”⟩) = (𝐹‘⟨“𝑣”⟩))
16 eqid 2621 . . . . . . . . . . . 12 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))
17 fvex 6163 . . . . . . . . . . . 12 (𝐹‘⟨“𝑣”⟩) ∈ V
1815, 16, 17fvmpt 6244 . . . . . . . . . . 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 4025 . . . . . . . . . . . . . . 15 ((𝐶𝑉) ∖ 𝑉) = (𝐶𝑉)
2120eleq2i 2690 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ 𝑣 ∈ (𝐶𝑉))
22 eldif 3569 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
2321, 22bitr3i 266 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐶𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
24 simpr2 1066 . . . . . . . . . . . . . 14 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
25 s1eq 13327 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑣 → ⟨“𝑐”⟩ = ⟨“𝑣”⟩)
2625fveq2d 6157 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑣 → (𝐹‘⟨“𝑐”⟩) = (𝐹‘⟨“𝑣”⟩))
2726, 25eqeq12d 2636 . . . . . . . . . . . . . . 15 (𝑐 = 𝑣 → ((𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ↔ (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩))
2827rspccva 3297 . . . . . . . . . . . . . 14 ((∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
2924, 28sylan 488 . . . . . . . . . . . . 13 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3023, 29sylan2br 493 . . . . . . . . . . . 12 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3130anassrs 679 . . . . . . . . . . 11 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3231eqcomd 2627 . . . . . . . . . 10 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → ⟨“𝑣”⟩ = (𝐹‘⟨“𝑣”⟩))
3319, 32ifeqda 4098 . . . . . . . . 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 4709 . . . . . . . 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 5248 . . . . . . 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 6626 . . . . . 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 4698 . . . . 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 3764 . . . . . . . 8 (𝑣𝑉𝑣 ∈ (𝐶𝑉))
39 simpr1 1065 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:𝑅𝑅)
4039adantr 481 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝐹:𝑅𝑅)
41 simpr 477 . . . . . . . . . . 11 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑣 ∈ (𝐶𝑉))
4241s1cld 13330 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ Word (𝐶𝑉))
4312ad2antrr 761 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑅 = Word (𝐶𝑉))
4442, 43eleqtrrd 2701 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ 𝑅)
4540, 44ffvelrnd 6321 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4638, 45sylan2 491 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4715cbvmptv 4715 . . . . . . 7 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑣𝑉 ↦ (𝐹‘⟨“𝑣”⟩))
4846, 47fmptd 6346 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅)
49 ssid 3608 . . . . . 6 𝑉𝑉
50 eqid 2621 . . . . . . 7 (freeMnd‘(𝐶𝑉)) = (freeMnd‘(𝐶𝑉))
515, 4, 2, 1, 50mrsubfval 31148 . . . . . 6 (((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) = (𝑟𝑅 ↦ ((freeMnd‘(𝐶𝑉)) Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣𝑉, ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑟))))
5248, 49, 51sylancl 693 . . . . 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))))
53 fvex 6163 . . . . . . . . . 10 (mCN‘𝑇) ∈ V
545, 53eqeltri 2694 . . . . . . . . 9 𝐶 ∈ V
55 fvex 6163 . . . . . . . . . 10 (mVR‘𝑇) ∈ V
564, 55eqeltri 2694 . . . . . . . . 9 𝑉 ∈ V
5754, 56unex 6916 . . . . . . . 8 (𝐶𝑉) ∈ V
5850frmdmnd 17328 . . . . . . . 8 ((𝐶𝑉) ∈ V → (freeMnd‘(𝐶𝑉)) ∈ Mnd)
5957, 58ax-mp 5 . . . . . . 7 (freeMnd‘(𝐶𝑉)) ∈ Mnd
6059a1i 11 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (freeMnd‘(𝐶𝑉)) ∈ Mnd)
6157a1i 11 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐶𝑉) ∈ V)
6245, 43eleqtrd 2700 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ Word (𝐶𝑉))
63 eqid 2621 . . . . . . 7 (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩))
6462, 63fmptd 6346 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)):(𝐶𝑉)⟶Word (𝐶𝑉))
6513, 13feq23d 6002 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹:𝑅𝑅𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉)))
6639, 65mpbid 222 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉))
67 simpr3 1067 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
68 simprl 793 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
6912adantr 481 . . . . . . . . . . . . . . . 16 ((𝑇𝑊𝐹:𝑅𝑅) → 𝑅 = Word (𝐶𝑉))
7069adantr 481 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑅 = Word (𝐶𝑉))
7168, 70eleqtrd 2700 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥 ∈ Word (𝐶𝑉))
72 simprr 795 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
7372, 70eleqtrd 2700 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦 ∈ Word (𝐶𝑉))
74 eqid 2621 . . . . . . . . . . . . . . . . . 18 (Base‘(freeMnd‘(𝐶𝑉))) = (Base‘(freeMnd‘(𝐶𝑉)))
7550, 74frmdbas 17321 . . . . . . . . . . . . . . . . 17 ((𝐶𝑉) ∈ V → (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉))
7657, 75ax-mp 5 . . . . . . . . . . . . . . . 16 (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉)
7776eqcomi 2630 . . . . . . . . . . . . . . 15 Word (𝐶𝑉) = (Base‘(freeMnd‘(𝐶𝑉)))
78 eqid 2621 . . . . . . . . . . . . . . 15 (+g‘(freeMnd‘(𝐶𝑉))) = (+g‘(freeMnd‘(𝐶𝑉)))
7950, 77, 78frmdadd 17324 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (𝐶𝑉) ∧ 𝑦 ∈ Word (𝐶𝑉)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
8071, 73, 79syl2anc 692 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
8180fveq2d 6157 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = (𝐹‘(𝑥 ++ 𝑦)))
82 ffvelrn 6318 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑥𝑅) → (𝐹𝑥) ∈ 𝑅)
8382ad2ant2lr 783 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ 𝑅)
8483, 70eleqtrd 2700 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ Word (𝐶𝑉))
85 ffvelrn 6318 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑦𝑅) → (𝐹𝑦) ∈ 𝑅)
8685ad2ant2l 781 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ 𝑅)
8786, 70eleqtrd 2700 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ Word (𝐶𝑉))
8850, 77, 78frmdadd 17324 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ Word (𝐶𝑉) ∧ (𝐹𝑦) ∈ Word (𝐶𝑉)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8984, 87, 88syl2anc 692 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
9081, 89eqeq12d 2636 . . . . . . . . . . 11 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
91902ralbidva 2983 . . . . . . . . . 10 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
9269raleqdv 3136 . . . . . . . . . . 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))))
9369, 92raleqbidv 3144 . . . . . . . . . 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))))
9491, 93bitr3d 270 . . . . . . . . 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))))
95943ad2antr1 1224 . . . . . . . 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))))
9667, 95mpbid 222 . . . . . . 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)))
97 wrd0 13277 . . . . . . . . . . . 12 ∅ ∈ Word (𝐶𝑉)
98 ffvelrn 6318 . . . . . . . . . . . 12 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ ∅ ∈ Word (𝐶𝑉)) → (𝐹‘∅) ∈ Word (𝐶𝑉))
9966, 97, 98sylancl 693 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) ∈ Word (𝐶𝑉))
100 lencl 13271 . . . . . . . . . . 11 ((𝐹‘∅) ∈ Word (𝐶𝑉) → (#‘(𝐹‘∅)) ∈ ℕ0)
10199, 100syl 17 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) ∈ ℕ0)
102101nn0cnd 11305 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) ∈ ℂ)
103 0cnd 9985 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 0 ∈ ℂ)
104102addid1d 10188 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((#‘(𝐹‘∅)) + 0) = (#‘(𝐹‘∅)))
10597, 13syl5eleqr 2705 . . . . . . . . . . . 12 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∅ ∈ 𝑅)
106 oveq1 6617 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝑥 ++ 𝑦) = (∅ ++ 𝑦))
107106fveq2d 6157 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (𝐹‘(𝑥 ++ 𝑦)) = (𝐹‘(∅ ++ 𝑦)))
108 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
109108oveq1d 6625 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝐹𝑥) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)))
110107, 109eqeq12d 2636 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ (𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦))))
111 oveq2 6618 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (∅ ++ 𝑦) = (∅ ++ ∅))
112 ccatlid 13316 . . . . . . . . . . . . . . . . 17 (∅ ∈ Word (𝐶𝑉) → (∅ ++ ∅) = ∅)
11397, 112ax-mp 5 . . . . . . . . . . . . . . . 16 (∅ ++ ∅) = ∅
114111, 113syl6eq 2671 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∅ ++ 𝑦) = ∅)
115114fveq2d 6157 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐹‘(∅ ++ 𝑦)) = (𝐹‘∅))
116 fveq2 6153 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
117116oveq2d 6626 . . . . . . . . . . . . . 14 (𝑦 = ∅ → ((𝐹‘∅) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹‘∅)))
118115, 117eqeq12d 2636 . . . . . . . . . . . . 13 (𝑦 = ∅ → ((𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)) ↔ (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅))))
119110, 118rspc2va 3311 . . . . . . . . . . . 12 (((∅ ∈ 𝑅 ∧ ∅ ∈ 𝑅) ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
120105, 105, 67, 119syl21anc 1322 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
121120fveq2d 6157 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) = (#‘((𝐹‘∅) ++ (𝐹‘∅))))
122 ccatlen 13307 . . . . . . . . . . 11 (((𝐹‘∅) ∈ Word (𝐶𝑉) ∧ (𝐹‘∅) ∈ Word (𝐶𝑉)) → (#‘((𝐹‘∅) ++ (𝐹‘∅))) = ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))))
12399, 99, 122syl2anc 692 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘((𝐹‘∅) ++ (𝐹‘∅))) = ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))))
124104, 121, 1233eqtrrd 2660 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))) = ((#‘(𝐹‘∅)) + 0))
125102, 102, 103, 124addcanad 10193 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) = 0)
126 fvex 6163 . . . . . . . . 9 (𝐹‘∅) ∈ V
127 hasheq0 13102 . . . . . . . . 9 ((𝐹‘∅) ∈ V → ((#‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅))
128126, 127ax-mp 5 . . . . . . . 8 ((#‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅)
129125, 128sylib 208 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ∅)
13059, 59pm3.2i 471 . . . . . . . 8 ((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (freeMnd‘(𝐶𝑉)) ∈ Mnd)
13150frmd0 17329 . . . . . . . . 9 ∅ = (0g‘(freeMnd‘(𝐶𝑉)))
13277, 77, 78, 78, 131, 131ismhm 17269 . . . . . . . 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‘∅) = ∅)))
133130, 132mpbiran 952 . . . . . . 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‘∅) = ∅))
13466, 96, 129, 133syl3anbrc 1244 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))))
135 eqid 2621 . . . . . . . . . 10 (varFMnd‘(𝐶𝑉)) = (varFMnd‘(𝐶𝑉))
136135vrmdf 17327 . . . . . . . . 9 ((𝐶𝑉) ∈ V → (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉))
13757, 136ax-mp 5 . . . . . . . 8 (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)
138 fcompt 6360 . . . . . . . 8 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))))
13966, 137, 138sylancl 693 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))))
140135vrmdval 17326 . . . . . . . . . 10 (((𝐶𝑉) ∈ V ∧ 𝑣 ∈ (𝐶𝑉)) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
14161, 140sylan 488 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
142141fveq2d 6157 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣)) = (𝐹‘⟨“𝑣”⟩))
143142mpteq2dva 4709 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)))
144139, 143eqtrd 2655 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)))
14550, 77, 135frmdup3lem 17335 . . . . . 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))))
14660, 61, 64, 134, 144, 145syl32anc 1331 . . . . 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))))
14737, 52, 1463eqtr4rd 2666 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 = (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))))
1484, 2, 1mrsubff 31152 . . . . . . 7 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
149 ffn 6007 . . . . . . 7 (𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅) → 𝑆 Fn (𝑅pm 𝑉))
150148, 149syl 17 . . . . . 6 (𝑇𝑊𝑆 Fn (𝑅pm 𝑉))
151150adantr 481 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑆 Fn (𝑅pm 𝑉))
152 fvex 6163 . . . . . . . 8 (mREx‘𝑇) ∈ V
1532, 152eqeltri 2694 . . . . . . 7 𝑅 ∈ V
154 elpm2r 7827 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉)) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
155153, 56, 154mpanl12 717 . . . . . 6 (((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
15648, 49, 155sylancl 693 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
157 fnfvelrn 6317 . . . . 5 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
158151, 156, 157syl2anc 692 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
159147, 158eqeltrd 2698 . . 3 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ran 𝑆)
160159ex 450 . 2 (𝑇𝑊 → ((𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → 𝐹 ∈ ran 𝑆))
16111, 160impbid2 216 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 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  Vcvv 3189  cdif 3556  cun 3557  wss 3559  c0 3896  ifcif 4063  cmpt 4678  ran crn 5080  ccom 5083   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809  pm cpm 7810  0cc0 9888   + caddc 9891  0cn0 11244  #chash 13065  Word cword 13238   ++ cconcat 13240  ⟨“cs1 13241  Basecbs 15792  +gcplusg 15873   Σg cgsu 16033  Mndcmnd 17226   MndHom cmhm 17265  freeMndcfrmd 17316  varFMndcvrmd 17317  mCNcmcn 31100  mVRcmvar 31101  mRExcmrex 31106  mRSubstcmrsub 31110
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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-card 8717  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-xnn0 11316  df-z 11330  df-uz 11640  df-fz 12277  df-fzo 12415  df-seq 12750  df-hash 13066  df-word 13246  df-lsw 13247  df-concat 13248  df-s1 13249  df-substr 13250  df-struct 15794  df-ndx 15795  df-slot 15796  df-base 15797  df-sets 15798  df-ress 15799  df-plusg 15886  df-0g 16034  df-gsum 16035  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-mhm 17267  df-submnd 17268  df-frmd 17318  df-vrmd 17319  df-mrex 31126  df-mrsub 31130
This theorem is referenced by:  mrsubco  31161
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