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Theorem elmrsubrn 33195
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 33224.) (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 33192 . . 3 (𝐹 ∈ ran 𝑆𝐹:𝑅𝑅)
4 mrsubcn.v . . . . 5 𝑉 = (mVR‘𝑇)
5 mrsubcn.c . . . . 5 𝐶 = (mCN‘𝑇)
61, 2, 4, 5mrsubcn 33194 . . . 4 ((𝐹 ∈ ran 𝑆𝑐 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
76ralrimiva 3105 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
81, 2mrsubccat 33193 . . . . 5 ((𝐹 ∈ ran 𝑆𝑥𝑅𝑦𝑅) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
983expb 1122 . . . 4 ((𝐹 ∈ ran 𝑆 ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
109ralrimivva 3112 . . 3 (𝐹 ∈ ran 𝑆 → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
113, 7, 103jca 1130 . 2 (𝐹 ∈ ran 𝑆 → (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
125, 4, 2mrexval 33176 . . . . . . 7 (𝑇𝑊𝑅 = Word (𝐶𝑉))
1312adantr 484 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑅 = Word (𝐶𝑉))
14 s1eq 14157 . . . . . . . . . . . . 13 (𝑤 = 𝑣 → ⟨“𝑤”⟩ = ⟨“𝑣”⟩)
1514fveq2d 6721 . . . . . . . . . . . 12 (𝑤 = 𝑣 → (𝐹‘⟨“𝑤”⟩) = (𝐹‘⟨“𝑣”⟩))
16 eqid 2737 . . . . . . . . . . . 12 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))
17 fvex 6730 . . . . . . . . . . . 12 (𝐹‘⟨“𝑣”⟩) ∈ V
1815, 16, 17fvmpt 6818 . . . . . . . . . . 11 (𝑣𝑉 → ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))‘𝑣) = (𝐹‘⟨“𝑣”⟩))
1918adantl 485 . . . . . . . . . 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 4395 . . . . . . . . . . . . . . 15 ((𝐶𝑉) ∖ 𝑉) = (𝐶𝑉)
2120eleq2i 2829 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ 𝑣 ∈ (𝐶𝑉))
22 eldif 3876 . . . . . . . . . . . . . 14 (𝑣 ∈ ((𝐶𝑉) ∖ 𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
2321, 22bitr3i 280 . . . . . . . . . . . . 13 (𝑣 ∈ (𝐶𝑉) ↔ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉))
24 simpr2 1197 . . . . . . . . . . . . . 14 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩)
25 s1eq 14157 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑣 → ⟨“𝑐”⟩ = ⟨“𝑣”⟩)
2625fveq2d 6721 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑣 → (𝐹‘⟨“𝑐”⟩) = (𝐹‘⟨“𝑣”⟩))
2726, 25eqeq12d 2753 . . . . . . . . . . . . . . 15 (𝑐 = 𝑣 → ((𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ↔ (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩))
2827rspccva 3536 . . . . . . . . . . . . . 14 ((∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
2924, 28sylan 583 . . . . . . . . . . . . 13 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3023, 29sylan2br 598 . . . . . . . . . . . 12 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ (𝑣 ∈ (𝐶𝑉) ∧ ¬ 𝑣𝑉)) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3130anassrs 471 . . . . . . . . . . 11 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) = ⟨“𝑣”⟩)
3231eqcomd 2743 . . . . . . . . . 10 ((((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) ∧ ¬ 𝑣𝑉) → ⟨“𝑣”⟩ = (𝐹‘⟨“𝑣”⟩))
3319, 32ifeqda 4475 . . . . . . . . 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 5150 . . . . . . . 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 5730 . . . . . . 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 7229 . . . . . 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 5140 . . . . 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 4091 . . . . . . . 8 (𝑣𝑉𝑣 ∈ (𝐶𝑉))
39 simplr1 1217 . . . . . . . . 9 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝐹:𝑅𝑅)
40 simpr 488 . . . . . . . . . . 11 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → 𝑣 ∈ (𝐶𝑉))
4140s1cld 14160 . . . . . . . . . 10 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → ⟨“𝑣”⟩ ∈ Word (𝐶𝑉))
4212ad2antrr 726 . . . . . . . . . 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, 43ffvelrnd 6905 . . . . . . . 8 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣 ∈ (𝐶𝑉)) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4538, 44sylan2 596 . . . . . . 7 (((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) ∧ 𝑣𝑉) → (𝐹‘⟨“𝑣”⟩) ∈ 𝑅)
4615cbvmptv 5158 . . . . . . 7 (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) = (𝑣𝑉 ↦ (𝐹‘⟨“𝑣”⟩))
4745, 46fmptd 6931 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅)
48 ssid 3923 . . . . . 6 𝑉𝑉
49 eqid 2737 . . . . . . 7 (freeMnd‘(𝐶𝑉)) = (freeMnd‘(𝐶𝑉))
505, 4, 2, 1, 49mrsubfval 33183 . . . . . 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 589 . . . . 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 6731 . . . . . . . . 9 𝐶 ∈ V
534fvexi 6731 . . . . . . . . 9 𝑉 ∈ V
5452, 53unex 7531 . . . . . . . 8 (𝐶𝑉) ∈ V
5549frmdmnd 18286 . . . . . . . 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 6932 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩)):(𝐶𝑉)⟶Word (𝐶𝑉))
61 simpr1 1196 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:𝑅𝑅)
6213, 13feq23d 6540 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹:𝑅𝑅𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉)))
6361, 62mpbid 235 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉))
64 simpr3 1198 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
65 simprl 771 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥𝑅)
6612adantr 484 . . . . . . . . . . . . . . . 16 ((𝑇𝑊𝐹:𝑅𝑅) → 𝑅 = Word (𝐶𝑉))
6766adantr 484 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑅 = Word (𝐶𝑉))
6865, 67eleqtrd 2840 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑥 ∈ Word (𝐶𝑉))
69 simprr 773 . . . . . . . . . . . . . . 15 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦𝑅)
7069, 67eleqtrd 2840 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → 𝑦 ∈ Word (𝐶𝑉))
71 eqid 2737 . . . . . . . . . . . . . . . . . 18 (Base‘(freeMnd‘(𝐶𝑉))) = (Base‘(freeMnd‘(𝐶𝑉)))
7249, 71frmdbas 18279 . . . . . . . . . . . . . . . . 17 ((𝐶𝑉) ∈ V → (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉))
7354, 72ax-mp 5 . . . . . . . . . . . . . . . 16 (Base‘(freeMnd‘(𝐶𝑉))) = Word (𝐶𝑉)
7473eqcomi 2746 . . . . . . . . . . . . . . 15 Word (𝐶𝑉) = (Base‘(freeMnd‘(𝐶𝑉)))
75 eqid 2737 . . . . . . . . . . . . . . 15 (+g‘(freeMnd‘(𝐶𝑉))) = (+g‘(freeMnd‘(𝐶𝑉)))
7649, 74, 75frmdadd 18282 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (𝐶𝑉) ∧ 𝑦 ∈ Word (𝐶𝑉)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
7768, 70, 76syl2anc 587 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦) = (𝑥 ++ 𝑦))
7877fveq2d 6721 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = (𝐹‘(𝑥 ++ 𝑦)))
79 ffvelrn 6902 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑥𝑅) → (𝐹𝑥) ∈ 𝑅)
8079ad2ant2lr 748 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ 𝑅)
8180, 67eleqtrd 2840 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑥) ∈ Word (𝐶𝑉))
82 ffvelrn 6902 . . . . . . . . . . . . . . 15 ((𝐹:𝑅𝑅𝑦𝑅) → (𝐹𝑦) ∈ 𝑅)
8382ad2ant2l 746 . . . . . . . . . . . . . 14 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ 𝑅)
8483, 67eleqtrd 2840 . . . . . . . . . . . . 13 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → (𝐹𝑦) ∈ Word (𝐶𝑉))
8549, 74, 75frmdadd 18282 . . . . . . . . . . . . 13 (((𝐹𝑥) ∈ Word (𝐶𝑉) ∧ (𝐹𝑦) ∈ Word (𝐶𝑉)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8681, 84, 85syl2anc 587 . . . . . . . . . . . 12 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))
8778, 86eqeq12d 2753 . . . . . . . . . . 11 (((𝑇𝑊𝐹:𝑅𝑅) ∧ (𝑥𝑅𝑦𝑅)) → ((𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
88872ralbidva 3119 . . . . . . . . . 10 ((𝑇𝑊𝐹:𝑅𝑅) → (∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥(+g‘(freeMnd‘(𝐶𝑉)))𝑦)) = ((𝐹𝑥)(+g‘(freeMnd‘(𝐶𝑉)))(𝐹𝑦)) ↔ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))))
8966raleqdv 3325 . . . . . . . . . . 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 3313 . . . . . . . . . 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 284 . . . . . . . . 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 1190 . . . . . . . 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 235 . . . . . . 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 14094 . . . . . . . . . . . 12 ∅ ∈ Word (𝐶𝑉)
95 ffvelrn 6902 . . . . . . . . . . . 12 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ ∅ ∈ Word (𝐶𝑉)) → (𝐹‘∅) ∈ Word (𝐶𝑉))
9663, 94, 95sylancl 589 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) ∈ Word (𝐶𝑉))
97 lencl 14088 . . . . . . . . . . 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 12152 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) ∈ ℂ)
100 0cnd 10826 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 0 ∈ ℂ)
10199addid1d 11032 . . . . . . . . . 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 7236 . . . . . . . . . . . . . 14 (𝑥 = ∅ → (𝐹‘(𝑥 ++ 𝑦)) = (𝐹‘(∅ ++ 𝑦)))
104 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑥 = ∅ → (𝐹𝑥) = (𝐹‘∅))
105104oveq1d 7228 . . . . . . . . . . . . . 14 (𝑥 = ∅ → ((𝐹𝑥) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)))
106103, 105eqeq12d 2753 . . . . . . . . . . . . 13 (𝑥 = ∅ → ((𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)) ↔ (𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦))))
107 oveq2 7221 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → (∅ ++ 𝑦) = (∅ ++ ∅))
108 ccatidid 14147 . . . . . . . . . . . . . . . 16 (∅ ++ ∅) = ∅
109107, 108eqtrdi 2794 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (∅ ++ 𝑦) = ∅)
110109fveq2d 6721 . . . . . . . . . . . . . 14 (𝑦 = ∅ → (𝐹‘(∅ ++ 𝑦)) = (𝐹‘∅))
111 fveq2 6717 . . . . . . . . . . . . . . 15 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
112111oveq2d 7229 . . . . . . . . . . . . . 14 (𝑦 = ∅ → ((𝐹‘∅) ++ (𝐹𝑦)) = ((𝐹‘∅) ++ (𝐹‘∅)))
113110, 112eqeq12d 2753 . . . . . . . . . . . . 13 (𝑦 = ∅ → ((𝐹‘(∅ ++ 𝑦)) = ((𝐹‘∅) ++ (𝐹𝑦)) ↔ (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅))))
114106, 113rspc2va 3548 . . . . . . . . . . . 12 (((∅ ∈ 𝑅 ∧ ∅ ∈ 𝑅) ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
115102, 102, 64, 114syl21anc 838 . . . . . . . . . . 11 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ((𝐹‘∅) ++ (𝐹‘∅)))
116115fveq2d 6721 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) = (♯‘((𝐹‘∅) ++ (𝐹‘∅))))
117 ccatlen 14130 . . . . . . . . . . 11 (((𝐹‘∅) ∈ Word (𝐶𝑉) ∧ (𝐹‘∅) ∈ Word (𝐶𝑉)) → (♯‘((𝐹‘∅) ++ (𝐹‘∅))) = ((♯‘(𝐹‘∅)) + (♯‘(𝐹‘∅))))
11896, 96, 117syl2anc 587 . . . . . . . . . 10 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘((𝐹‘∅) ++ (𝐹‘∅))) = ((♯‘(𝐹‘∅)) + (♯‘(𝐹‘∅))))
119101, 116, 1183eqtrrd 2782 . . . . . . . . 9 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → ((♯‘(𝐹‘∅)) + (♯‘(𝐹‘∅))) = ((♯‘(𝐹‘∅)) + 0))
12099, 99, 100, 119addcanad 11037 . . . . . . . 8 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (♯‘(𝐹‘∅)) = 0)
121 fvex 6730 . . . . . . . . 9 (𝐹‘∅) ∈ V
122 hasheq0 13930 . . . . . . . . 9 ((𝐹‘∅) ∈ V → ((♯‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅))
123121, 122ax-mp 5 . . . . . . . 8 ((♯‘(𝐹‘∅)) = 0 ↔ (𝐹‘∅) = ∅)
124120, 123sylib 221 . . . . . . 7 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝐹‘∅) = ∅)
12556, 56pm3.2i 474 . . . . . . . 8 ((freeMnd‘(𝐶𝑉)) ∈ Mnd ∧ (freeMnd‘(𝐶𝑉)) ∈ Mnd)
12649frmd0 18287 . . . . . . . . 9 ∅ = (0g‘(freeMnd‘(𝐶𝑉)))
12774, 74, 75, 75, 126, 126ismhm 18220 . . . . . . . 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 709 . . . . . . 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 1345 . . . . . 6 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 ∈ ((freeMnd‘(𝐶𝑉)) MndHom (freeMnd‘(𝐶𝑉))))
130 eqid 2737 . . . . . . . . . 10 (varFMnd‘(𝐶𝑉)) = (varFMnd‘(𝐶𝑉))
131130vrmdf 18285 . . . . . . . . 9 ((𝐶𝑉) ∈ V → (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉))
13254, 131ax-mp 5 . . . . . . . 8 (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)
133 fcompt 6948 . . . . . . . 8 ((𝐹:Word (𝐶𝑉)⟶Word (𝐶𝑉) ∧ (varFMnd‘(𝐶𝑉)):(𝐶𝑉)⟶Word (𝐶𝑉)) → (𝐹 ∘ (varFMnd‘(𝐶𝑉))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))))
13463, 132, 133sylancl 589 . . . . . . 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 18284 . . . . . . . . . 10 (((𝐶𝑉) ∈ V ∧ 𝑣 ∈ (𝐶𝑉)) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
13654, 135mpan 690 . . . . . . . . 9 (𝑣 ∈ (𝐶𝑉) → ((varFMnd‘(𝐶𝑉))‘𝑣) = ⟨“𝑣”⟩)
137136fveq2d 6721 . . . . . . . 8 (𝑣 ∈ (𝐶𝑉) → (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣)) = (𝐹‘⟨“𝑣”⟩))
138137mpteq2ia 5146 . . . . . . 7 (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘((varFMnd‘(𝐶𝑉))‘𝑣))) = (𝑣 ∈ (𝐶𝑉) ↦ (𝐹‘⟨“𝑣”⟩))
139134, 138eqtrdi 2794 . . . . . 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 18293 . . . . . 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 1380 . . . . 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 2788 . . . 4 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝐹 = (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))))
1434, 2, 1mrsubff 33187 . . . . . . 7 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
144143ffnd 6546 . . . . . 6 (𝑇𝑊𝑆 Fn (𝑅pm 𝑉))
145144adantr 484 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → 𝑆 Fn (𝑅pm 𝑉))
1462fvexi 6731 . . . . . . 7 𝑅 ∈ V
147 elpm2r 8526 . . . . . . 7 (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ ((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉)) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
148146, 53, 147mpanl12 702 . . . . . 6 (((𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)):𝑉𝑅𝑉𝑉) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
14947, 48, 148sylancl 589 . . . . 5 ((𝑇𝑊 ∧ (𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦)))) → (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉))
150 fnfvelrn 6901 . . . . 5 ((𝑆 Fn (𝑅pm 𝑉) ∧ (𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩)) ∈ (𝑅pm 𝑉)) → (𝑆‘(𝑤𝑉 ↦ (𝐹‘⟨“𝑤”⟩))) ∈ ran 𝑆)
151145, 149, 150syl2anc 587 . . . 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 416 . 2 (𝑇𝑊 → ((𝐹:𝑅𝑅 ∧ ∀𝑐 ∈ (𝐶𝑉)(𝐹‘⟨“𝑐”⟩) = ⟨“𝑐”⟩ ∧ ∀𝑥𝑅𝑦𝑅 (𝐹‘(𝑥 ++ 𝑦)) = ((𝐹𝑥) ++ (𝐹𝑦))) → 𝐹 ∈ ran 𝑆))
15411, 153impbid2 229 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  Vcvv 3408  cdif 3863  cun 3864  wss 3866  c0 4237  ifcif 4439  cmpt 5135  ran crn 5552  ccom 5555   Fn wfn 6375  wf 6376  cfv 6380  (class class class)co 7213  m cmap 8508  pm cpm 8509  0cc0 10729   + caddc 10732  0cn0 12090  chash 13896  Word cword 14069   ++ cconcat 14125  ⟨“cs1 14152  Basecbs 16760  +gcplusg 16802   Σg cgsu 16945  Mndcmnd 18173   MndHom cmhm 18216  freeMndcfrmd 18274  varFMndcvrmd 18275  mCNcmcn 33135  mVRcmvar 33136  mRExcmrex 33141  mRSubstcmrsub 33145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-0g 16946  df-gsum 16947  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-frmd 18276  df-vrmd 18277  df-mrex 33161  df-mrsub 33165
This theorem is referenced by:  mrsubco  33196
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