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Theorem elmrsubrn 30671
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 30700.) (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 30668 . . 3 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
4 mrsubcn.v . . . . 5 𝑉 = (mVR‘𝑇)
5 mrsubcn.c . . . . 5 𝐶 = (mCN‘𝑇)
61, 2, 4, 5mrsubcn 30670 . . . 4 ((𝐹 ∈ ran 𝑆𝑐 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
76ralrimiva 2949 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
81, 2mrsubccat 30669 . . . . 5 ((𝐹 ∈ ran 𝑆𝑥𝑅𝑦𝑅) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
983expb 1258 . . . 4 ((𝐹 ∈ ran 𝑆 ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
109ralrimivva 2954 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
113, 7, 103jca 1235 . 2 (𝐹 ∈ ran 𝑆 → (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
125, 4, 2mrexval 30652 . . . . . . 7 (𝑇𝑊𝑅 = Word (𝐶𝑉))
1312adantr 480 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑅 = Word (𝐶𝑉))
14 s1eq 13233 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → ⟨“𝑤”⟩ = ⟨“𝑣”⟩)
1514fveq2d 6107 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝐹‘⟨“𝑤”⟩) = (𝐹‘⟨“𝑣”⟩))
16 eqid 2610 . . . . . . . . . . . 12 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))
17 fvex 6113 . . . . . . . . . . . 12 (𝐹‘⟨“𝑣”⟩) ∈ V
1815, 16, 17fvmpt 6191 . . . . . . . . . . 11 (𝑣𝑉 → ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣) = (𝐹‘⟨“𝑣”⟩))
1918adantl 481 . . . . . . . . . 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 4000 . . . . . . . . . . . . . . 15 ((𝐶𝑉) ∖ 𝑉) = (𝐶𝑉)
2120eleq2i 2680 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ 𝑣 ∈ (𝐶𝑉))
22 eldif 3550 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
2321, 22bitr3i 265 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐶𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
24 simpr2 1061 . . . . . . . . . . . . . 14 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
25 s1eq 13233 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑣 → ⟨“𝑐”⟩ = ⟨“𝑣”⟩)
2625fveq2d 6107 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑣 → (𝐹‘⟨“𝑐”⟩) = (𝐹‘⟨“𝑣”⟩))
2726, 25eqeq12d 2625 . . . . . . . . . . . . . . 15 (𝑐 = 𝑣 → ((𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ↔ (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩))
2827rspccva 3281 . . . . . . . . . . . . . 14 ((∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
2924, 28sylan 487 . . . . . . . . . . . . 13 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3023, 29sylan2br 492 . . . . . . . . . . . 12 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3130anassrs 678 . . . . . . . . . . 11 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3231eqcomd 2616 . . . . . . . . . 10 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → ⟨“𝑣”⟩ = (𝐹‘⟨“𝑣”⟩))
3319, 32ifeqda 4071 . . . . . . . . 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 4672 . . . . . . . 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 5205 . . . . . . 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 6565 . . . . . 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 4663 . . . . 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 3743 . . . . . . . 8 (𝑣𝑉𝑣 ∈ (𝐶𝑉))
39 simpr1 1060 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:𝑅𝑅)
4039adantr 480 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝐹:𝑅𝑅)
41 simpr 476 . . . . . . . . . . 11 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑣 ∈ (𝐶𝑉))
4241s1cld 13236 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ Word (𝐶𝑉))
4312ad2antrr 758 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑅 = Word (𝐶𝑉))
4442, 43eleqtrrd 2691 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ 𝑅)
4540, 44ffvelrnd 6268 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4638, 45sylan2 490 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4715cbvmptv 4678 . . . . . . 7 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑣𝑉 ↦ (𝐹‘⟨“𝑣”⟩))
4846, 47fmptd 6292 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅)
49 ssid 3587 . . . . . 6 𝑉𝑉
50 eqid 2610 . . . . . . 7 (freeMnd‘(𝐶𝑉)) = (freeMnd‘(𝐶𝑉))
515, 4, 2, 1, 50mrsubfval 30659 . . . . . 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 6113 . . . . . . . . . 10 (mCN‘𝑇) ∈ V
545, 53eqeltri 2684 . . . . . . . . 9 𝐶 ∈ V
55 fvex 6113 . . . . . . . . . 10 (mVR‘𝑇) ∈ V
564, 55eqeltri 2684 . . . . . . . . 9 𝑉 ∈ V
5754, 56unex 6854 . . . . . . . 8 (𝐶𝑉) ∈ V
5850frmdmnd 17219 . . . . . . . 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 2690 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ Word (𝐶𝑉))
63 eqid 2610 . . . . . . 7 (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩))
6462, 63fmptd 6292 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)):(𝐶𝑉)⟶Word (𝐶𝑉))
6513, 13feq23d 5953 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹:𝑅𝑅𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉)))
6639, 65mpbid 221 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉))
67 simpr3 1062 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
68 simprl 790 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
6912adantr 480 . . . . . . . . . . . . . . . 16 ((𝑇𝑊𝐹:𝑅𝑅) → 𝑅 = Word (𝐶𝑉))
7069adantr 480 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑅 = Word (𝐶𝑉))
7168, 70eleqtrd 2690 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥 ∈ Word (𝐶𝑉))
72 simprr 792 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
7372, 70eleqtrd 2690 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦 ∈ Word (𝐶𝑉))
74 eqid 2610 . . . . . . . . . . . . . . . . . 18 (Base‘(freeMnd‘(𝐶𝑉))) = (Base‘(freeMnd‘(𝐶𝑉)))
7550, 74frmdbas 17212 . . . . . . . . . . . . . . . . 17 ((𝐶𝑉) ∈ V → (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉))
7657, 75ax-mp 5 . . . . . . . . . . . . . . . 16 (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉)
7776eqcomi 2619 . . . . . . . . . . . . . . 15 Word (𝐶𝑉) = (Base‘(freeMnd‘(𝐶𝑉)))
78 eqid 2610 . . . . . . . . . . . . . . 15 (+g‘(freeMnd‘(𝐶𝑉))) = (+g‘(freeMnd‘(𝐶𝑉)))
7950, 77, 78frmdadd 17215 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (𝐶𝑉) ∧ 𝑦 ∈ Word (𝐶𝑉)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
8071, 73, 79syl2anc 691 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
8180fveq2d 6107 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = (𝐹‘(𝑥 ++ 𝑦)))
82 ffvelrn 6265 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑥𝑅) → (𝐹𝑥) ∈ 𝑅)
8382ad2ant2lr 780 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ 𝑅)
8483, 70eleqtrd 2690 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ Word (𝐶𝑉))
85 ffvelrn 6265 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑦𝑅) → (𝐹𝑦) ∈ 𝑅)
8685ad2ant2l 778 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ 𝑅)
8786, 70eleqtrd 2690 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ Word (𝐶𝑉))
8850, 77, 78frmdadd 17215 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ Word (𝐶𝑉) ∧ (𝐹𝑦) ∈ Word (𝐶𝑉)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8984, 87, 88syl2anc 691 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
9081, 89eqeq12d 2625 . . . . . . . . . . 11 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
91902ralbidva 2971 . . . . . . . . . 10 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
9269raleqdv 3121 . . . . . . . . . . 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 3129 . . . . . . . . . 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 269 . . . . . . . . 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 1219 . . . . . . . 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 221 . . . . . . 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 13185 . . . . . . . . . . . 12 ∅ ∈ Word (𝐶𝑉)
98 ffvelrn 6265 . . . . . . . . . . . 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 13179 . . . . . . . . . . 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 11230 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) ∈ ℂ)
103 0cnd 9912 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 0 ∈ ℂ)
104102addid1d 10115 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((#‘(𝐹‘∅)) + 0) = (#‘(𝐹‘∅)))
10597, 13syl5eleqr 2695 . . . . . . . . . . . 12 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∅ ∈ 𝑅)
106 oveq1 6556 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝑥 ++ 𝑦) = (∅ ++ 𝑦))
107106fveq2d 6107 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (𝐹‘(𝑥 ++ 𝑦)) = (𝐹‘(∅ ++ 𝑦)))
108 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
109108oveq1d 6564 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝐹𝑥) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)))
110107, 109eqeq12d 2625 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ (𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦))))
111 oveq2 6557 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (∅ ++ 𝑦) = (∅ ++ ∅))
112 ccatlid 13222 . . . . . . . . . . . . . . . . 17 (∅ ∈ Word (𝐶𝑉) → (∅ ++ ∅) = ∅)
11397, 112ax-mp 5 . . . . . . . . . . . . . . . 16 (∅ ++ ∅) = ∅
114111, 113syl6eq 2660 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∅ ++ 𝑦) = ∅)
115114fveq2d 6107 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐹‘(∅ ++ 𝑦)) = (𝐹‘∅))
116 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
117116oveq2d 6565 . . . . . . . . . . . . . 14 (𝑦 = ∅ → ((𝐹‘∅) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹‘∅)))
118115, 117eqeq12d 2625 . . . . . . . . . . . . 13 (𝑦 = ∅ → ((𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)) ↔ (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅))))
119110, 118rspc2va 3294 . . . . . . . . . . . 12 (((∅ ∈ 𝑅 ∧ ∅ ∈ 𝑅) ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
120105, 105, 67, 119syl21anc 1317 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
121120fveq2d 6107 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) = (#‘((𝐹‘∅) ++ (𝐹‘∅))))
122 ccatlen 13213 . . . . . . . . . . 11 (((𝐹‘∅) ∈ Word (𝐶𝑉) ∧ (𝐹‘∅) ∈ Word (𝐶𝑉)) → (#‘((𝐹‘∅) ++ (𝐹‘∅))) = ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))))
12399, 99, 122syl2anc 691 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘((𝐹‘∅) ++ (𝐹‘∅))) = ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))))
124104, 121, 1233eqtrrd 2649 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))) = ((#‘(𝐹‘∅)) + 0))
125102, 102, 103, 124addcanad 10120 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) = 0)
126 fvex 6113 . . . . . . . . 9 (𝐹‘∅) ∈ V
127 hasheq0 13015 . . . . . . . . 9 ((𝐹‘∅) ∈ V → ((#‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅))
128126, 127ax-mp 5 . . . . . . . 8 ((#‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅)
129125, 128sylib 207 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ∅)
13059, 59pm3.2i 470 . . . . . . . 8 ((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (freeMnd‘(𝐶𝑉)) ∈ Mnd)
13150frmd0 17220 . . . . . . . . 9 ∅ = (0g‘(freeMnd‘(𝐶𝑉)))
13277, 77, 78, 78, 131, 131ismhm 17160 . . . . . . . 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 955 . . . . . . 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 1239 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))))
135 eqid 2610 . . . . . . . . . 10 (varFMnd‘(𝐶𝑉)) = (varFMnd‘(𝐶𝑉))
136135vrmdf 17218 . . . . . . . . 9 ((𝐶𝑉) ∈ V → (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉))
13757, 136ax-mp 5 . . . . . . . 8 (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)
138 fcompt 6306 . . . . . . . 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 17217 . . . . . . . . . 10 (((𝐶𝑉) ∈ V ∧ 𝑣 ∈ (𝐶𝑉)) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
14161, 140sylan 487 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
142141fveq2d 6107 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣)) = (𝐹‘⟨“𝑣”⟩))
143142mpteq2dva 4672 . . . . . . 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 2644 . . . . . 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 17226 . . . . . 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 1326 . . . . 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 2655 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 = (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))))
1484, 2, 1mrsubff 30663 . . . . . . 7 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
149 ffn 5958 . . . . . . 7 (𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅) → 𝑆 Fn (𝑅pm 𝑉))
150148, 149syl 17 . . . . . 6 (𝑇𝑊𝑆 Fn (𝑅pm 𝑉))
151150adantr 480 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑆 Fn (𝑅pm 𝑉))
152 fvex 6113 . . . . . . . 8 (mREx‘𝑇) ∈ V
1532, 152eqeltri 2684 . . . . . . 7 𝑅 ∈ V
154 elpm2r 7761 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉)) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
155153, 56, 154mpanl12 714 . . . . . 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 6264 . . . . 5 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
158151, 156, 157syl2anc 691 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
159147, 158eqeltrd 2688 . . 3 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ran 𝑆)
160159ex 449 . 2 (𝑇𝑊 → ((𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → 𝐹 ∈ ran 𝑆))
16111, 160impbid2 215 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 195  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  Vcvv 3173  cdif 3537  cun 3538  wss 3540  c0 3874  ifcif 4036  cmpt 4643  ran crn 5039  ccom 5042   Fn wfn 5799  wf 5800  cfv 5804  (class class class)co 6549  𝑚 cmap 7744  pm cpm 7745  0cc0 9815   + caddc 9818  0cn0 11169  #chash 12979  Word cword 13146   ++ cconcat 13148  ⟨“cs1 13149  Basecbs 15695  +gcplusg 15768   Σg cgsu 15924  Mndcmnd 17117   MndHom cmhm 17156  freeMndcfrmd 17207  varFMndcvrmd 17208  mCNcmcn 30611  mVRcmvar 30612  mRExcmrex 30617  mRSubstcmrsub 30621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-frmd 17209  df-vrmd 17210  df-mrex 30637  df-mrsub 30641
This theorem is referenced by:  mrsubco  30672
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