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Theorem mrsubccat 35523
Description: Substitution distributes over concatenation. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubccat.s 𝑆 = (mRSubst‘𝑇)
mrsubccat.r 𝑅 = (mREx‘𝑇)
Assertion
Ref Expression
mrsubccat ((𝐹 ∈ ran 𝑆𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))

Proof of Theorem mrsubccat
Dummy variables 𝑓 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4340 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubccat.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6900 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 141 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2737 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
6 mrsubccat.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6, 2mrsubff 35517 . . . . 5 (𝑇 ∈ V → 𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅))
8 ffun 6739 . . . . 5 (𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅) → Fun 𝑆)
94, 7, 83syl 18 . . . 4 (𝐹 ∈ ran 𝑆 → Fun 𝑆)
105, 6, 2mrsubrn 35518 . . . . . 6 ran 𝑆 = (𝑆 “ (𝑅m (mVR‘𝑇)))
1110eleq2i 2833 . . . . 5 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
1211biimpi 216 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
13 fvelima 6974 . . . 4 ((Fun 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇)))) → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
149, 12, 13syl2anc 584 . . 3 (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
15 simprl 771 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋𝑅)
16 elfvex 6944 . . . . . . . . . . . . . 14 (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
1716, 6eleq2s 2859 . . . . . . . . . . . . 13 (𝑋𝑅𝑇 ∈ V)
18 eqid 2737 . . . . . . . . . . . . . 14 (mCN‘𝑇) = (mCN‘𝑇)
1918, 5, 6mrexval 35506 . . . . . . . . . . . . 13 (𝑇 ∈ V → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2015, 17, 193syl 18 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2115, 20eleqtrd 2843 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22 simprr 773 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌𝑅)
2322, 20eleqtrd 2843 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24 elmapi 8889 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝑅m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2524adantr 480 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2625adantr 480 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶𝑅)
2726ffvelcdmda 7104 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓𝑣) ∈ 𝑅)
2820ad2antrr 726 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2927, 28eleqtrd 2843 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
30 simplr 769 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3130s1cld 14641 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3229, 31ifclda 4561 . . . . . . . . . . . 12 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3332fmpttd 7135 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
34 ccatco 14874 . . . . . . . . . . 11 ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌)) = (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
3521, 23, 33, 34syl3anc 1373 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌)) = (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
3635oveq2d 7447 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
37 fvex 6919 . . . . . . . . . . . 12 (mCN‘𝑇) ∈ V
38 fvex 6919 . . . . . . . . . . . 12 (mVR‘𝑇) ∈ V
3937, 38unex 7764 . . . . . . . . . . 11 ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40 eqid 2737 . . . . . . . . . . . 12 (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
4140frmdmnd 18872 . . . . . . . . . . 11 (((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V → (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd)
4239, 41mp1i 13 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd)
43 wrdco 14870 . . . . . . . . . . 11 ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
4421, 33, 43syl2anc 584 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
45 wrdco 14870 . . . . . . . . . . 11 ((𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ (mVR‘𝑇))⟶Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
4623, 33, 45syl2anc 584 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
47 eqid 2737 . . . . . . . . . . . . . 14 (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
4840, 47frmdbas 18865 . . . . . . . . . . . . 13 (((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V → (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
4939, 48ax-mp 5 . . . . . . . . . . . 12 (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = Word ((mCN‘𝑇) ∪ (mVR‘𝑇))
5049eqcomi 2746 . . . . . . . . . . 11 Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51 eqid 2737 . . . . . . . . . . 11 (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
5250, 51gsumccat 18854 . . . . . . . . . 10 (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
5342, 44, 46, 52syl3anc 1373 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg (((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ++ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
5450gsumwcl 18852 . . . . . . . . . . 11 (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5542, 44, 54syl2anc 584 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5650gsumwcl 18852 . . . . . . . . . . 11 (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌) ∈ Word Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5742, 46, 56syl2anc 584 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
5840, 50, 51frmdadd 18868 . . . . . . . . . 10 ((((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
5955, 57, 58syl2anc 584 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋))(+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
6036, 53, 593eqtrd 2781 . . . . . . . 8 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
61 ssidd 4007 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62 ccatcl 14612 . . . . . . . . . . 11 ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6321, 23, 62syl2anc 584 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6463, 20eleqtrrd 2844 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
6518, 5, 6, 2, 40mrsubval 35514 . . . . . . . . 9 ((𝑓:(mVR‘𝑇)⟶𝑅 ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ (𝑋 ++ 𝑌) ∈ 𝑅) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))))
6625, 61, 64, 65syl3anc 1373 . . . . . . . 8 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ (𝑋 ++ 𝑌))))
6718, 5, 6, 2, 40mrsubval 35514 . . . . . . . . . 10 ((𝑓:(mVR‘𝑇)⟶𝑅 ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ 𝑋𝑅) → ((𝑆𝑓)‘𝑋) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
6825, 61, 15, 67syl3anc 1373 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘𝑋) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))
6918, 5, 6, 2, 40mrsubval 35514 . . . . . . . . . 10 ((𝑓:(mVR‘𝑇)⟶𝑅 ∧ (mVR‘𝑇) ⊆ (mVR‘𝑇) ∧ 𝑌𝑅) → ((𝑆𝑓)‘𝑌) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
7025, 61, 22, 69syl3anc 1373 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘𝑌) = ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌)))
7168, 70oveq12d 7449 . . . . . . . 8 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) = (((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)) ++ ((freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)) ↦ if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑌))))
7260, 66, 713eqtr4d 2787 . . . . . . 7 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)))
73 fveq1 6905 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74 fveq1 6905 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑋) = (𝐹𝑋))
75 fveq1 6905 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑌) = (𝐹𝑌))
7674, 75oveq12d 7449 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
7773, 76eqeq12d 2753 . . . . . . 7 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) ↔ (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
7872, 77syl5ibcom 245 . . . . . 6 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
7978ex 412 . . . . 5 (𝑓 ∈ (𝑅m (mVR‘𝑇)) → ((𝑋𝑅𝑌𝑅) → ((𝑆𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))))
8079com23 86 . . . 4 (𝑓 ∈ (𝑅m (mVR‘𝑇)) → ((𝑆𝑓) = 𝐹 → ((𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))))
8180rexlimiv 3148 . . 3 (∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹 → ((𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
8214, 81syl 17 . 2 (𝐹 ∈ ran 𝑆 → ((𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
83823impib 1117 1 ((𝐹 ∈ ran 𝑆𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  cun 3949  wss 3951  c0 4333  ifcif 4525  cmpt 5225  ran crn 5686  cima 5688  ccom 5689  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  pm cpm 8867  Word cword 14552   ++ cconcat 14608  ⟨“cs1 14633  Basecbs 17247  +gcplusg 17297   Σg cgsu 17485  Mndcmnd 18747  freeMndcfrmd 18860  mCNcmcn 35465  mVRcmvar 35466  mRExcmrex 35471  mRSubstcmrsub 35475
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-concat 14609  df-s1 14634  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-frmd 18862  df-mrex 35491  df-mrsub 35495
This theorem is referenced by:  elmrsubrn  35525  mrsubco  35526  mrsubvrs  35527
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