Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mrsubccat Structured version   Visualization version   GIF version

Theorem mrsubccat 35037
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 4328 . . . . . 6 (𝐹 ∈ ran 𝑆 → ¬ ran 𝑆 = ∅)
2 mrsubccat.s . . . . . . 7 𝑆 = (mRSubst‘𝑇)
32rnfvprc 6879 . . . . . 6 𝑇 ∈ V → ran 𝑆 = ∅)
41, 3nsyl2 141 . . . . 5 (𝐹 ∈ ran 𝑆𝑇 ∈ V)
5 eqid 2726 . . . . . 6 (mVR‘𝑇) = (mVR‘𝑇)
6 mrsubccat.r . . . . . 6 𝑅 = (mREx‘𝑇)
75, 6, 2mrsubff 35031 . . . . 5 (𝑇 ∈ V → 𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅))
8 ffun 6714 . . . . 5 (𝑆:(𝑅pm (mVR‘𝑇))⟶(𝑅m 𝑅) → Fun 𝑆)
94, 7, 83syl 18 . . . 4 (𝐹 ∈ ran 𝑆 → Fun 𝑆)
105, 6, 2mrsubrn 35032 . . . . . 6 ran 𝑆 = (𝑆 “ (𝑅m (mVR‘𝑇)))
1110eleq2i 2819 . . . . 5 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
1211biimpi 215 . . . 4 (𝐹 ∈ ran 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇))))
13 fvelima 6951 . . . 4 ((Fun 𝑆𝐹 ∈ (𝑆 “ (𝑅m (mVR‘𝑇)))) → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
149, 12, 13syl2anc 583 . . 3 (𝐹 ∈ ran 𝑆 → ∃𝑓 ∈ (𝑅m (mVR‘𝑇))(𝑆𝑓) = 𝐹)
15 simprl 768 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋𝑅)
16 elfvex 6923 . . . . . . . . . . . . . 14 (𝑋 ∈ (mREx‘𝑇) → 𝑇 ∈ V)
1716, 6eleq2s 2845 . . . . . . . . . . . . 13 (𝑋𝑅𝑇 ∈ V)
18 eqid 2726 . . . . . . . . . . . . . 14 (mCN‘𝑇) = (mCN‘𝑇)
1918, 5, 6mrexval 35020 . . . . . . . . . . . . 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 2829 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
22 simprr 770 . . . . . . . . . . . 12 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌𝑅)
2322, 20eleqtrd 2829 . . . . . . . . . . 11 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
24 elmapi 8845 . . . . . . . . . . . . . . . . 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 7080 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓𝑣) ∈ 𝑅)
2820ad2antrr 723 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → 𝑅 = Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
2927, 28eleqtrd 2829 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ 𝑣 ∈ (mVR‘𝑇)) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
30 simplr 766 . . . . . . . . . . . . . 14 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3130s1cld 14559 . . . . . . . . . . . . 13 ((((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) ∧ ¬ 𝑣 ∈ (mVR‘𝑇)) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3229, 31ifclda 4558 . . . . . . . . . . . 12 (((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ (mVR‘𝑇))) → if(𝑣 ∈ (mVR‘𝑇), (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
3332fmpttd 7110 . . . . . . . . . . 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 14792 . . . . . . . . . . 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 7421 . . . . . . . . 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 6898 . . . . . . . . . . . 12 (mCN‘𝑇) ∈ V
38 fvex 6898 . . . . . . . . . . . 12 (mVR‘𝑇) ∈ V
3937, 38unex 7730 . . . . . . . . . . 11 ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∈ V
40 eqid 2726 . . . . . . . . . . . 12 (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))) = (freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))
4140frmdmnd 18784 . . . . . . . . . . 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 14788 . . . . . . . . . . 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 583 . . . . . . . . . 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 14788 . . . . . . . . . . 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 583 . . . . . . . . . 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 2726 . . . . . . . . . . . . . 14 (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
4840, 47frmdbas 18777 . . . . . . . . . . . . 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 2735 . . . . . . . . . . 11 Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
51 eqid 2726 . . . . . . . . . . 11 (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇)))) = (+g‘(freeMnd‘((mCN‘𝑇) ∪ (mVR‘𝑇))))
5250, 51gsumccat 18766 . . . . . . . . . 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 18764 . . . . . . . . . . 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 583 . . . . . . . . . 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 18764 . . . . . . . . . . 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 583 . . . . . . . . . 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 18780 . . . . . . . . . 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 583 . . . . . . . . 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 2770 . . . . . . . 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 4000 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (mVR‘𝑇) ⊆ (mVR‘𝑇))
62 ccatcl 14530 . . . . . . . . . . 11 ((𝑋 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)) ∧ 𝑌 ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇))) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6321, 23, 62syl2anc 583 . . . . . . . . . 10 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ Word ((mCN‘𝑇) ∪ (mVR‘𝑇)))
6463, 20eleqtrrd 2830 . . . . . . . . 9 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → (𝑋 ++ 𝑌) ∈ 𝑅)
6518, 5, 6, 2, 40mrsubval 35028 . . . . . . . . 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 35028 . . . . . . . . . 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 35028 . . . . . . . . . 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 7423 . . . . . . . 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 2776 . . . . . . 7 ((𝑓 ∈ (𝑅m (mVR‘𝑇)) ∧ (𝑋𝑅𝑌𝑅)) → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)))
73 fveq1 6884 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘(𝑋 ++ 𝑌)) = (𝐹‘(𝑋 ++ 𝑌)))
74 fveq1 6884 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑋) = (𝐹𝑋))
75 fveq1 6884 . . . . . . . . 9 ((𝑆𝑓) = 𝐹 → ((𝑆𝑓)‘𝑌) = (𝐹𝑌))
7674, 75oveq12d 7423 . . . . . . . 8 ((𝑆𝑓) = 𝐹 → (((𝑆𝑓)‘𝑋) ++ ((𝑆𝑓)‘𝑌)) = ((𝐹𝑋) ++ (𝐹𝑌)))
7773, 76eqeq12d 2742 . . . . . . 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 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 3142 . . 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 395  w3a 1084   = wceq 1533  wcel 2098  wrex 3064  Vcvv 3468  cun 3941  wss 3943  c0 4317  ifcif 4523  cmpt 5224  ran crn 5670  cima 5672  ccom 5673  Fun wfun 6531  wf 6533  cfv 6537  (class class class)co 7405  m cmap 8822  pm cpm 8823  Word cword 14470   ++ cconcat 14526  ⟨“cs1 14551  Basecbs 17153  +gcplusg 17206   Σg cgsu 17395  Mndcmnd 18667  freeMndcfrmd 18772  mCNcmcn 34979  mVRcmvar 34980  mRExcmrex 34985  mRSubstcmrsub 34989
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-1o 8467  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14296  df-word 14471  df-concat 14527  df-s1 14552  df-struct 17089  df-sets 17106  df-slot 17124  df-ndx 17136  df-base 17154  df-ress 17183  df-plusg 17219  df-0g 17396  df-gsum 17397  df-mgm 18573  df-sgrp 18652  df-mnd 18668  df-submnd 18714  df-frmd 18774  df-mrex 35005  df-mrsub 35009
This theorem is referenced by:  elmrsubrn  35039  mrsubco  35040  mrsubvrs  35041
  Copyright terms: Public domain W3C validator