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Theorem elmrsubrn 31543
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 31572.) (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 31540 . . 3 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
4 mrsubcn.v . . . . 5 𝑉 = (mVR‘𝑇)
5 mrsubcn.c . . . . 5 𝐶 = (mCN‘𝑇)
61, 2, 4, 5mrsubcn 31542 . . . 4 ((𝐹 ∈ ran 𝑆𝑐 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
76ralrimiva 2995 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
81, 2mrsubccat 31541 . . . . 5 ((𝐹 ∈ ran 𝑆𝑥𝑅𝑦𝑅) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
983expb 1285 . . . 4 ((𝐹 ∈ ran 𝑆 ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
109ralrimivva 3000 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
113, 7, 103jca 1261 . 2 (𝐹 ∈ ran 𝑆 → (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
125, 4, 2mrexval 31524 . . . . . . 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 13416 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → ⟨“𝑤”⟩ = ⟨“𝑣”⟩)
1514fveq2d 6233 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝐹‘⟨“𝑤”⟩) = (𝐹‘⟨“𝑣”⟩))
16 eqid 2651 . . . . . . . . . . . 12 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))
17 fvex 6239 . . . . . . . . . . . 12 (𝐹‘⟨“𝑣”⟩) ∈ V
1815, 16, 17fvmpt 6321 . . . . . . . . . . 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 4081 . . . . . . . . . . . . . . 15 ((𝐶𝑉) ∖ 𝑉) = (𝐶𝑉)
2120eleq2i 2722 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ 𝑣 ∈ (𝐶𝑉))
22 eldif 3617 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
2321, 22bitr3i 266 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐶𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
24 simpr2 1088 . . . . . . . . . . . . . 14 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
25 s1eq 13416 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑣 → ⟨“𝑐”⟩ = ⟨“𝑣”⟩)
2625fveq2d 6233 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑣 → (𝐹‘⟨“𝑐”⟩) = (𝐹‘⟨“𝑣”⟩))
2726, 25eqeq12d 2666 . . . . . . . . . . . . . . 15 (𝑐 = 𝑣 → ((𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ↔ (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩))
2827rspccva 3339 . . . . . . . . . . . . . 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 681 . . . . . . . . . . 11 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3231eqcomd 2657 . . . . . . . . . 10 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → ⟨“𝑣”⟩ = (𝐹‘⟨“𝑣”⟩))
3319, 32ifeqda 4154 . . . . . . . . 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 4777 . . . . . . . 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 5316 . . . . . . 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 6706 . . . . . 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 4766 . . . . 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 3814 . . . . . . . 8 (𝑣𝑉𝑣 ∈ (𝐶𝑉))
39 simpr1 1087 . . . . . . . . . 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 13419 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ Word (𝐶𝑉))
4312ad2antrr 762 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑅 = Word (𝐶𝑉))
4442, 43eleqtrrd 2733 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ 𝑅)
4540, 44ffvelrnd 6400 . . . . . . . 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 4783 . . . . . . 7 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑣𝑉 ↦ (𝐹‘⟨“𝑣”⟩))
4846, 47fmptd 6425 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅)
49 ssid 3657 . . . . . 6 𝑉𝑉
50 eqid 2651 . . . . . . 7 (freeMnd‘(𝐶𝑉)) = (freeMnd‘(𝐶𝑉))
515, 4, 2, 1, 50mrsubfval 31531 . . . . . 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 695 . . . . 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 6239 . . . . . . . . . 10 (mCN‘𝑇) ∈ V
545, 53eqeltri 2726 . . . . . . . . 9 𝐶 ∈ V
55 fvex 6239 . . . . . . . . . 10 (mVR‘𝑇) ∈ V
564, 55eqeltri 2726 . . . . . . . . 9 𝑉 ∈ V
5754, 56unex 6998 . . . . . . . 8 (𝐶𝑉) ∈ V
5850frmdmnd 17443 . . . . . . . 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 2732 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ Word (𝐶𝑉))
63 eqid 2651 . . . . . . 7 (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩))
6462, 63fmptd 6425 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)):(𝐶𝑉)⟶Word (𝐶𝑉))
6513, 13feq23d 6078 . . . . . . . 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 1089 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
68 simprl 809 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
6912adantr 480 . . . . . . . . . . . . . . . 16 ((𝑇𝑊𝐹:𝑅𝑅) → 𝑅 = Word (𝐶𝑉))
7069adantr 480 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑅 = Word (𝐶𝑉))
7168, 70eleqtrd 2732 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥 ∈ Word (𝐶𝑉))
72 simprr 811 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
7372, 70eleqtrd 2732 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦 ∈ Word (𝐶𝑉))
74 eqid 2651 . . . . . . . . . . . . . . . . . 18 (Base‘(freeMnd‘(𝐶𝑉))) = (Base‘(freeMnd‘(𝐶𝑉)))
7550, 74frmdbas 17436 . . . . . . . . . . . . . . . . 17 ((𝐶𝑉) ∈ V → (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉))
7657, 75ax-mp 5 . . . . . . . . . . . . . . . 16 (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉)
7776eqcomi 2660 . . . . . . . . . . . . . . 15 Word (𝐶𝑉) = (Base‘(freeMnd‘(𝐶𝑉)))
78 eqid 2651 . . . . . . . . . . . . . . 15 (+g‘(freeMnd‘(𝐶𝑉))) = (+g‘(freeMnd‘(𝐶𝑉)))
7950, 77, 78frmdadd 17439 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (𝐶𝑉) ∧ 𝑦 ∈ Word (𝐶𝑉)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
8071, 73, 79syl2anc 694 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
8180fveq2d 6233 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = (𝐹‘(𝑥 ++ 𝑦)))
82 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑥𝑅) → (𝐹𝑥) ∈ 𝑅)
8382ad2ant2lr 799 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ 𝑅)
8483, 70eleqtrd 2732 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ Word (𝐶𝑉))
85 ffvelrn 6397 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑦𝑅) → (𝐹𝑦) ∈ 𝑅)
8685ad2ant2l 797 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ 𝑅)
8786, 70eleqtrd 2732 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ Word (𝐶𝑉))
8850, 77, 78frmdadd 17439 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ Word (𝐶𝑉) ∧ (𝐹𝑦) ∈ Word (𝐶𝑉)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8984, 87, 88syl2anc 694 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
9081, 89eqeq12d 2666 . . . . . . . . . . 11 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
91902ralbidva 3017 . . . . . . . . . 10 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
9269raleqdv 3174 . . . . . . . . . . 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 3182 . . . . . . . . . 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 1246 . . . . . . . 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 13362 . . . . . . . . . . . 12 ∅ ∈ Word (𝐶𝑉)
98 ffvelrn 6397 . . . . . . . . . . . 12 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ ∅ ∈ Word (𝐶𝑉)) → (𝐹‘∅) ∈ Word (𝐶𝑉))
9966, 97, 98sylancl 695 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) ∈ Word (𝐶𝑉))
100 lencl 13356 . . . . . . . . . . 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 11391 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) ∈ ℂ)
103 0cnd 10071 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 0 ∈ ℂ)
104102addid1d 10274 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((#‘(𝐹‘∅)) + 0) = (#‘(𝐹‘∅)))
10597, 13syl5eleqr 2737 . . . . . . . . . . . 12 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∅ ∈ 𝑅)
106 oveq1 6697 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝑥 ++ 𝑦) = (∅ ++ 𝑦))
107106fveq2d 6233 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (𝐹‘(𝑥 ++ 𝑦)) = (𝐹‘(∅ ++ 𝑦)))
108 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
109108oveq1d 6705 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝐹𝑥) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)))
110107, 109eqeq12d 2666 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ (𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦))))
111 oveq2 6698 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (∅ ++ 𝑦) = (∅ ++ ∅))
112 ccatlid 13404 . . . . . . . . . . . . . . . . 17 (∅ ∈ Word (𝐶𝑉) → (∅ ++ ∅) = ∅)
11397, 112ax-mp 5 . . . . . . . . . . . . . . . 16 (∅ ++ ∅) = ∅
114111, 113syl6eq 2701 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∅ ++ 𝑦) = ∅)
115114fveq2d 6233 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐹‘(∅ ++ 𝑦)) = (𝐹‘∅))
116 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
117116oveq2d 6706 . . . . . . . . . . . . . 14 (𝑦 = ∅ → ((𝐹‘∅) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹‘∅)))
118115, 117eqeq12d 2666 . . . . . . . . . . . . 13 (𝑦 = ∅ → ((𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)) ↔ (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅))))
119110, 118rspc2va 3354 . . . . . . . . . . . 12 (((∅ ∈ 𝑅 ∧ ∅ ∈ 𝑅) ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
120105, 105, 67, 119syl21anc 1365 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
121120fveq2d 6233 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) = (#‘((𝐹‘∅) ++ (𝐹‘∅))))
122 ccatlen 13393 . . . . . . . . . . 11 (((𝐹‘∅) ∈ Word (𝐶𝑉) ∧ (𝐹‘∅) ∈ Word (𝐶𝑉)) → (#‘((𝐹‘∅) ++ (𝐹‘∅))) = ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))))
12399, 99, 122syl2anc 694 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘((𝐹‘∅) ++ (𝐹‘∅))) = ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))))
124104, 121, 1233eqtrrd 2690 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((#‘(𝐹‘∅)) + (#‘(𝐹‘∅))) = ((#‘(𝐹‘∅)) + 0))
125102, 102, 103, 124addcanad 10279 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (#‘(𝐹‘∅)) = 0)
126 fvex 6239 . . . . . . . . 9 (𝐹‘∅) ∈ V
127 hasheq0 13192 . . . . . . . . 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 470 . . . . . . . 8 ((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (freeMnd‘(𝐶𝑉)) ∈ Mnd)
13150frmd0 17444 . . . . . . . . 9 ∅ = (0g‘(freeMnd‘(𝐶𝑉)))
13277, 77, 78, 78, 131, 131ismhm 17384 . . . . . . . 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 973 . . . . . . 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 1265 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))))
135 eqid 2651 . . . . . . . . . 10 (varFMnd‘(𝐶𝑉)) = (varFMnd‘(𝐶𝑉))
136135vrmdf 17442 . . . . . . . . 9 ((𝐶𝑉) ∈ V → (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉))
13757, 136ax-mp 5 . . . . . . . 8 (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)
138 fcompt 6440 . . . . . . . 8 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))))
13966, 137, 138sylancl 695 . . . . . . 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 17441 . . . . . . . . . 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 6233 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣)) = (𝐹‘⟨“𝑣”⟩))
143142mpteq2dva 4777 . . . . . . 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 2685 . . . . . 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 17450 . . . . . 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 1374 . . . . 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 2696 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 = (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))))
1484, 2, 1mrsubff 31535 . . . . . . 7 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅𝑚 𝑅))
149 ffn 6083 . . . . . . 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 6239 . . . . . . . 8 (mREx‘𝑇) ∈ V
1532, 152eqeltri 2726 . . . . . . 7 𝑅 ∈ V
154 elpm2r 7917 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉)) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
155153, 56, 154mpanl12 718 . . . . . 6 (((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
15648, 49, 155sylancl 695 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
157 fnfvelrn 6396 . . . . 5 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
158151, 156, 157syl2anc 694 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
159147, 158eqeltrd 2730 . . 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 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 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  ifcif 4119  cmpt 4762  ran crn 5144  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  pm cpm 7900  0cc0 9974   + caddc 9977  0cn0 11330  #chash 13157  Word cword 13323   ++ cconcat 13325  ⟨“cs1 13326  Basecbs 15904  +gcplusg 15988   Σg cgsu 16148  Mndcmnd 17341   MndHom cmhm 17380  freeMndcfrmd 17431  varFMndcvrmd 17432  mCNcmcn 31483  mVRcmvar 31484  mRExcmrex 31489  mRSubstcmrsub 31493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-word 13331  df-lsw 13332  df-concat 13333  df-s1 13334  df-substr 13335  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-0g 16149  df-gsum 16150  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-frmd 17433  df-vrmd 17434  df-mrex 31509  df-mrsub 31513
This theorem is referenced by:  mrsubco  31544
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