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Theorem splval 13483
Description: Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
splval ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))

Proof of Theorem splval
Dummy variables 𝑠 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-splice 13287 . . 3 splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)))
21a1i 11 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩))))
3 simprl 793 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → 𝑠 = 𝑆)
4 fveq2 6178 . . . . . . . . 9 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st𝑏) = (1st ‘⟨𝐹, 𝑇, 𝑅⟩))
54fveq2d 6182 . . . . . . . 8 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
65adantl 482 . . . . . . 7 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘(1st𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
7 ot1stg 7167 . . . . . . . 8 ((𝐹𝑊𝑇𝑋𝑅𝑌) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
87adantl 482 . . . . . . 7 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
96, 8sylan9eqr 2676 . . . . . 6 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (1st ‘(1st𝑏)) = 𝐹)
109opeq2d 4400 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ⟨0, (1st ‘(1st𝑏))⟩ = ⟨0, 𝐹⟩)
113, 10oveq12d 6653 . . . 4 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) = (𝑆 substr ⟨0, 𝐹⟩))
12 fveq2 6178 . . . . . 6 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
1312adantl 482 . . . . 5 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
14 ot3rdg 7169 . . . . . . 7 (𝑅𝑌 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
15143ad2ant3 1082 . . . . . 6 ((𝐹𝑊𝑇𝑋𝑅𝑌) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
1615adantl 482 . . . . 5 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
1713, 16sylan9eqr 2676 . . . 4 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd𝑏) = 𝑅)
1811, 17oveq12d 6653 . . 3 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) = ((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅))
194fveq2d 6182 . . . . . . 7 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘(1st𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
2019adantl 482 . . . . . 6 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
21 ot2ndg 7168 . . . . . . 7 ((𝐹𝑊𝑇𝑋𝑅𝑌) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
2221adantl 482 . . . . . 6 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
2320, 22sylan9eqr 2676 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘(1st𝑏)) = 𝑇)
243fveq2d 6182 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (#‘𝑠) = (#‘𝑆))
2523, 24opeq12d 4401 . . . 4 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩ = ⟨𝑇, (#‘𝑆)⟩)
263, 25oveq12d 6653 . . 3 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩) = (𝑆 substr ⟨𝑇, (#‘𝑆)⟩))
2718, 26oveq12d 6653 . 2 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (((𝑠 substr ⟨0, (1st ‘(1st𝑏))⟩) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (#‘𝑠)⟩)) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
28 elex 3207 . . 3 (𝑆𝑉𝑆 ∈ V)
2928adantr 481 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → 𝑆 ∈ V)
30 otex 4924 . . 3 𝐹, 𝑇, 𝑅⟩ ∈ V
3130a1i 11 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
32 ovexd 6665 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)) ∈ V)
332, 27, 29, 31, 32ovmpt2d 6773 1 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 substr ⟨0, 𝐹⟩) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (#‘𝑆)⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174  cotp 4176  cfv 5876  (class class class)co 6635  cmpt2 6637  1st c1st 7151  2nd c2nd 7152  0cc0 9921  #chash 13100   ++ cconcat 13276   substr csubstr 13278   splice csplice 13279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-ot 4177  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-splice 13287
This theorem is referenced by:  splid  13485  spllen  13486  splfv1  13487  splfv2a  13488  splval2  13489  gsumspl  17362  efgredleme  18137  efgredlemc  18139  efgcpbllemb  18149  frgpuplem  18166  splvalpfx  41200
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