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Theorem splvalpfx 41206
Description: Value of the substring replacement operator. (Contributed by AV, 11-May-2020.)
Assertion
Ref Expression
splvalpfx ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))

Proof of Theorem splvalpfx
StepHypRef Expression
1 splval 13496 . 2 ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
2 pfxval 41154 . . . . . 6 ((𝑆𝑉𝐹 ∈ ℕ0) → (𝑆 prefix 𝐹) = (𝑆 substr ⟨0, 𝐹⟩))
323ad2antr1 1225 . . . . 5 ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (𝑆 prefix 𝐹) = (𝑆 substr ⟨0, 𝐹⟩))
43eqcomd 2627 . . . 4 ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (𝑆 substr ⟨0, 𝐹⟩) = (𝑆 prefix 𝐹))
54oveq1d 6662 . . 3 ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → ((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) = ((𝑆 prefix 𝐹) ++ 𝑅))
65oveq1d 6662 . 2 ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
71, 6eqtrd 2655 1 ((𝑆𝑉 ∧ (𝐹 ∈ ℕ0𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  cop 4181  cotp 4183  cfv 5886  (class class class)co 6647  0cc0 9933  0cn0 11289  #chash 13112   ++ cconcat 13288   substr csubstr 13290   splice csplice 13291   prefix cpfx 41152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-ot 4184  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-1st 7165  df-2nd 7166  df-splice 13299  df-pfx 41153
This theorem is referenced by: (None)
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