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Theorem mrsubccat 34112
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 4293 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubccat.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6836 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 141 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2736 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
6 mrsubccat.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6, 2mrsubff 34106 . . . . 5 (𝑇 ∈ V → 𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅))
8 ffun 6671 . . . . 5 (𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅) → Fun 𝑆)
94, 7, 83syl 18 . . . 4 (𝐹 ∈ ran 𝑆 → Fun 𝑆)
105, 6, 2mrsubrn 34107 . . . . . 6 ran 𝑆 = (𝑆 “ (𝑅m (mVR‘𝑇)))
1110eleq2i 2829 . . . . 5 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
1211biimpi 215 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
13 fvelima 6908 . . . 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 769 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋𝑅)
16 elfvex 6880 . . . . . . . . . . . . . 14 (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
1716, 6eleq2s 2856 . . . . . . . . . . . . 13 (𝑋𝑅𝑇 ∈ V)
18 eqid 2736 . . . . . . . . . . . . . 14 (mCN‘𝑇) = (mCN‘𝑇)
1918, 5, 6mrexval 34095 . . . . . . . . . . . . 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 2840 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22 simprr 771 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌𝑅)
2322, 20eleqtrd 2840 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24 elmapi 8787 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ (𝑅m (mVR‘𝑇)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2524adantr 481 . . . . . . . . . . . . . . . 16 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑓:(mVR‘𝑇)⟶𝑅)
2625adantr 481 . . . . . . . . . . . . . . 15 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → 𝑓:(mVR‘𝑇)⟶𝑅)
2726ffvelcdmda 7035 . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . 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 14491 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3229, 31ifclda 4521 . . . . . . . . . . . 12 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3332fmpttd 7063 . . . . . . . . . . 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 14724 . . . . . . . . . . 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 1371 . . . . . . . . . 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 7373 . . . . . . . . 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 6855 . . . . . . . . . . . 12 (mCN‘𝑇) ∈ V
38 fvex 6855 . . . . . . . . . . . 12 (mVR‘𝑇) ∈ V
3937, 38unex 7680 . . . . . . . . . . 11 ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40 eqid 2736 . . . . . . . . . . . 12 (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
4140frmdmnd 18669 . . . . . . . . . . 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 14720 . . . . . . . . . . 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 14720 . . . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
4840, 47frmdbas 18662 . . . . . . . . . . . . 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 2745 . . . . . . . . . . 11 Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51 eqid 2736 . . . . . . . . . . 11 (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
5250, 51gsumccat 18651 . . . . . . . . . 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 1371 . . . . . . . . 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 18649 . . . . . . . . . . 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 18649 . . . . . . . . . . 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 18665 . . . . . . . . . 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 2780 . . . . . . . 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 3967 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62 ccatcl 14462 . . . . . . . . . . 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 2841 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
6518, 5, 6, 2, 40mrsubval 34103 . . . . . . . . 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 1371 . . . . . . . 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 34103 . . . . . . . . . 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 1371 . . . . . . . . 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 34103 . . . . . . . . . 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 1371 . . . . . . . . 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 7375 . . . . . . . 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 2786 . . . . . . 7 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)))
73 fveq1 6841 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74 fveq1 6841 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑋) = (𝐹𝑋))
75 fveq1 6841 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑌) = (𝐹𝑌))
7674, 75oveq12d 7375 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
7773, 76eqeq12d 2752 . . . . . . 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 413 . . . . 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 1116 1 ((𝐹 ∈ ran 𝑆𝑋𝑅𝑌𝑅) → (𝐹‘(𝑋 ++ 𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  Vcvv 3445  cun 3908  wss 3910  c0 4282  ifcif 4486  cmpt 5188  ran crn 5634  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765  pm cpm 8766  Word cword 14402   ++ cconcat 14458  ⟨“cs1 14483  Basecbs 17083  +gcplusg 17133   Σg cgsu 17322  Mndcmnd 18556  freeMndcfrmd 18657  mCNcmcn 34054  mVRcmvar 34055  mRExcmrex 34060  mRSubstcmrsub 34064
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-word 14403  df-concat 14459  df-s1 14484  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-0g 17323  df-gsum 17324  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-frmd 18659  df-mrex 34080  df-mrsub 34084
This theorem is referenced by:  elmrsubrn  34114  mrsubco  34115  mrsubvrs  34116
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