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Theorem mrsubccat 35169
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 4337 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubccat.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6896 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 141 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2728 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
6 mrsubccat.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6, 2mrsubff 35163 . . . . 5 (𝑇 ∈ V → 𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅))
8 ffun 6730 . . . . 5 (𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅) → Fun 𝑆)
94, 7, 83syl 18 . . . 4 (𝐹 ∈ ran 𝑆 → Fun 𝑆)
105, 6, 2mrsubrn 35164 . . . . . 6 ran 𝑆 = (𝑆 “ (𝑅m (mVR‘𝑇)))
1110eleq2i 2821 . . . . 5 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
1211biimpi 215 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
13 fvelima 6969 . . . 4 ((Fun 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇)))) → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
149, 12, 13syl2anc 582 . . 3 (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
15 simprl 769 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋𝑅)
16 elfvex 6940 . . . . . . . . . . . . . 14 (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
1716, 6eleq2s 2847 . . . . . . . . . . . . 13 (𝑋𝑅𝑇 ∈ V)
18 eqid 2728 . . . . . . . . . . . . . 14 (mCN‘𝑇) = (mCN‘𝑇)
1918, 5, 6mrexval 35152 . . . . . . . . . . . . 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 2831 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22 simprr 771 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌𝑅)
2322, 20eleqtrd 2831 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24 elmapi 8876 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝑅m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2524adantr 479 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2625adantr 479 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶𝑅)
2726ffvelcdmda 7099 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓𝑣) ∈ 𝑅)
2820ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2927, 28eleqtrd 2831 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
30 simplr 767 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3130s1cld 14595 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3229, 31ifclda 4567 . . . . . . . . . . . 12 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3332fmpttd 7130 . . . . . . . . . . 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 14828 . . . . . . . . . . 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 1368 . . . . . . . . . 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 7442 . . . . . . . . 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 6915 . . . . . . . . . . . 12 (mCN‘𝑇) ∈ V
38 fvex 6915 . . . . . . . . . . . 12 (mVR‘𝑇) ∈ V
3937, 38unex 7756 . . . . . . . . . . 11 ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40 eqid 2728 . . . . . . . . . . . 12 (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
4140frmdmnd 18825 . . . . . . . . . . 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 14824 . . . . . . . . . . 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 582 . . . . . . . . . 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 14824 . . . . . . . . . . 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 582 . . . . . . . . . 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 2728 . . . . . . . . . . . . . 14 (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
4840, 47frmdbas 18818 . . . . . . . . . . . . 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 2737 . . . . . . . . . . 11 Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51 eqid 2728 . . . . . . . . . . 11 (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
5250, 51gsumccat 18807 . . . . . . . . . 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 1368 . . . . . . . . 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 18805 . . . . . . . . . . 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 582 . . . . . . . . . 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 18805 . . . . . . . . . . 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 582 . . . . . . . . . 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 18821 . . . . . . . . . 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 582 . . . . . . . . 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 2772 . . . . . . . 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 4005 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62 ccatcl 14566 . . . . . . . . . . 11 ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6321, 23, 62syl2anc 582 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6463, 20eleqtrrd 2832 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
6518, 5, 6, 2, 40mrsubval 35160 . . . . . . . . 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 1368 . . . . . . . 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 35160 . . . . . . . . . 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 1368 . . . . . . . . 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 35160 . . . . . . . . . 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 1368 . . . . . . . . 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 7444 . . . . . . . 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 2778 . . . . . . 7 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)))
73 fveq1 6901 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74 fveq1 6901 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑋) = (𝐹𝑋))
75 fveq1 6901 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑌) = (𝐹𝑌))
7674, 75oveq12d 7444 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
7773, 76eqeq12d 2744 . . . . . . 7 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) ↔ (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
7872, 77syl5ibcom 244 . . . . . 6 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓) = 𝐹 → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌))))
7978ex 411 . . . . 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 3145 . . 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 1113 1 ((𝐹 ∈ ran 𝑆𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  wrex 3067  Vcvv 3473  cun 3947  wss 3949  c0 4326  ifcif 4532  cmpt 5235  ran crn 5683  cima 5685  ccom 5686  Fun wfun 6547  wf 6549  cfv 6553  (class class class)co 7426  m cmap 8853  pm cpm 8854  Word cword 14506   ++ cconcat 14562  ⟨“cs1 14587  Basecbs 17189  +gcplusg 17242   Σg cgsu 17431  Mndcmnd 18703  freeMndcfrmd 18813  mCNcmcn 35111  mVRcmvar 35112  mRExcmrex 35117  mRSubstcmrsub 35121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748  ax-cnex 11204  ax-resscn 11205  ax-1cn 11206  ax-icn 11207  ax-addcl 11208  ax-addrcl 11209  ax-mulcl 11210  ax-mulrcl 11211  ax-mulcom 11212  ax-addass 11213  ax-mulass 11214  ax-distr 11215  ax-i2m1 11216  ax-1ne0 11217  ax-1rid 11218  ax-rnegex 11219  ax-rrecex 11220  ax-cnre 11221  ax-pre-lttri 11222  ax-pre-lttrn 11223  ax-pre-ltadd 11224  ax-pre-mulgt0 11225
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7879  df-1st 8001  df-2nd 8002  df-frecs 8295  df-wrecs 8326  df-recs 8400  df-rdg 8439  df-1o 8495  df-er 8733  df-map 8855  df-pm 8856  df-en 8973  df-dom 8974  df-sdom 8975  df-fin 8976  df-card 9972  df-pnf 11290  df-mnf 11291  df-xr 11292  df-ltxr 11293  df-le 11294  df-sub 11486  df-neg 11487  df-nn 12253  df-2 12315  df-n0 12513  df-z 12599  df-uz 12863  df-fz 13527  df-fzo 13670  df-seq 14009  df-hash 14332  df-word 14507  df-concat 14563  df-s1 14588  df-struct 17125  df-sets 17142  df-slot 17160  df-ndx 17172  df-base 17190  df-ress 17219  df-plusg 17255  df-0g 17432  df-gsum 17433  df-mgm 18609  df-sgrp 18688  df-mnd 18704  df-submnd 18750  df-frmd 18815  df-mrex 35137  df-mrsub 35141
This theorem is referenced by:  elmrsubrn  35171  mrsubco  35172  mrsubvrs  35173
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