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Theorem mrsubccat 33043
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 4220 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubccat.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6662 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 143 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2738 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
6 mrsubccat.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6, 2mrsubff 33037 . . . . 5 (𝑇 ∈ V → 𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅))
8 ffun 6501 . . . . 5 (𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅) → Fun 𝑆)
94, 7, 83syl 18 . . . 4 (𝐹 ∈ ran 𝑆 → Fun 𝑆)
105, 6, 2mrsubrn 33038 . . . . . 6 ran 𝑆 = (𝑆 “ (𝑅m (mVR‘𝑇)))
1110eleq2i 2824 . . . . 5 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
1211biimpi 219 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
13 fvelima 6729 . . . 4 ((Fun 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇)))) → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
149, 12, 13syl2anc 587 . . 3 (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
15 simprl 771 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋𝑅)
16 elfvex 6701 . . . . . . . . . . . . . 14 (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
1716, 6eleq2s 2851 . . . . . . . . . . . . 13 (𝑋𝑅𝑇 ∈ V)
18 eqid 2738 . . . . . . . . . . . . . 14 (mCN‘𝑇) = (mCN‘𝑇)
1918, 5, 6mrexval 33026 . . . . . . . . . . . . 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 2835 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22 simprr 773 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌𝑅)
2322, 20eleqtrd 2835 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24 elmapi 8452 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝑅m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2524adantr 484 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2625adantr 484 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶𝑅)
2726ffvelrnda 6855 . . . . . . . . . . . . . 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 2835 . . . . . . . . . . . . 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 14039 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3229, 31ifclda 4446 . . . . . . . . . . . 12 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3332fmpttd 6883 . . . . . . . . . . 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 14279 . . . . . . . . . . 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 1372 . . . . . . . . . 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 7180 . . . . . . . . 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 6681 . . . . . . . . . . . 12 (mCN‘𝑇) ∈ V
38 fvex 6681 . . . . . . . . . . . 12 (mVR‘𝑇) ∈ V
3937, 38unex 7481 . . . . . . . . . . 11 ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40 eqid 2738 . . . . . . . . . . . 12 (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
4140frmdmnd 18133 . . . . . . . . . . 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 14275 . . . . . . . . . . 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 587 . . . . . . . . . 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 14275 . . . . . . . . . . 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 587 . . . . . . . . . 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 2738 . . . . . . . . . . . . . 14 (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
4840, 47frmdbas 18126 . . . . . . . . . . . . 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 2747 . . . . . . . . . . 11 Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51 eqid 2738 . . . . . . . . . . 11 (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
5250, 51gsumccat 18115 . . . . . . . . . 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 1372 . . . . . . . . 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 18112 . . . . . . . . . . 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 587 . . . . . . . . . 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 18112 . . . . . . . . . . 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 587 . . . . . . . . . 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 18129 . . . . . . . . . 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 587 . . . . . . . . 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 2777 . . . . . . . 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 3898 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62 ccatcl 14008 . . . . . . . . . . 11 ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6321, 23, 62syl2anc 587 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6463, 20eleqtrrd 2836 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
6518, 5, 6, 2, 40mrsubval 33034 . . . . . . . . 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 1372 . . . . . . . 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 33034 . . . . . . . . . 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 1372 . . . . . . . . 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 33034 . . . . . . . . . 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 1372 . . . . . . . . 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 7182 . . . . . . . 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 2783 . . . . . . 7 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)))
73 fveq1 6667 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74 fveq1 6667 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑋) = (𝐹𝑋))
75 fveq1 6667 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑌) = (𝐹𝑌))
7674, 75oveq12d 7182 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
7773, 76eqeq12d 2754 . . . . . . 7 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) ↔ (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
7872, 77syl5ibcom 248 . . . . . 6 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
7978ex 416 . . . . 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 3189 . . 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 399  w3a 1088   = wceq 1542  wcel 2113  wrex 3054  Vcvv 3397  cun 3839  wss 3841  c0 4209  ifcif 4411  cmpt 5107  ran crn 5520  cima 5522  ccom 5523  Fun wfun 6327  wf 6329  cfv 6333  (class class class)co 7164  m cmap 8430  pm cpm 8431  Word cword 13948   ++ cconcat 14004  ⟨“cs1 14031  Basecbs 16579  +gcplusg 16661   Σg cgsu 16810  Mndcmnd 18020  freeMndcfrmd 18121  mCNcmcn 32985  mVRcmvar 32986  mRExcmrex 32991  mRSubstcmrsub 32995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-pm 8433  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-card 9434  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-n0 11970  df-z 12056  df-uz 12318  df-fz 12975  df-fzo 13118  df-seq 13454  df-hash 13776  df-word 13949  df-concat 14005  df-s1 14032  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-sets 16586  df-ress 16587  df-plusg 16674  df-0g 16811  df-gsum 16812  df-mgm 17961  df-sgrp 18010  df-mnd 18021  df-submnd 18066  df-frmd 18123  df-mrex 33011  df-mrsub 33015
This theorem is referenced by:  elmrsubrn  33045  mrsubco  33046  mrsubvrs  33047
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