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Theorem splval 14646
Description: Value of the substring replacement operator. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by AV, 11-May-2020.) (Revised by AV, 15-Oct-2022.)
Assertion
Ref Expression
splval ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 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 14645 . . 3 splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st𝑏))) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (♯‘𝑠)⟩)))
21a1i 11 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → splice = (𝑠 ∈ V, 𝑏 ∈ V ↦ (((𝑠 prefix (1st ‘(1st𝑏))) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (♯‘𝑠)⟩))))
3 simprl 770 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → 𝑠 = 𝑆)
4 2fveq3 6852 . . . . . . 7 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (1st ‘(1st𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
54adantl 483 . . . . . 6 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (1st ‘(1st𝑏)) = (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
6 ot1stg 7940 . . . . . . 7 ((𝐹𝑊𝑇𝑋𝑅𝑌) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
76adantl 483 . . . . . 6 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (1st ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝐹)
85, 7sylan9eqr 2799 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (1st ‘(1st𝑏)) = 𝐹)
93, 8oveq12d 7380 . . . 4 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 prefix (1st ‘(1st𝑏))) = (𝑆 prefix 𝐹))
10 fveq2 6847 . . . . . 6 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
1110adantl 483 . . . . 5 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝑇, 𝑅⟩))
12 ot3rdg 7942 . . . . . . 7 (𝑅𝑌 → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
13123ad2ant3 1136 . . . . . 6 ((𝐹𝑊𝑇𝑋𝑅𝑌) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
1413adantl 483 . . . . 5 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (2nd ‘⟨𝐹, 𝑇, 𝑅⟩) = 𝑅)
1511, 14sylan9eqr 2799 . . . 4 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd𝑏) = 𝑅)
169, 15oveq12d 7380 . . 3 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ((𝑠 prefix (1st ‘(1st𝑏))) ++ (2nd𝑏)) = ((𝑆 prefix 𝐹) ++ 𝑅))
17 2fveq3 6852 . . . . . . 7 (𝑏 = ⟨𝐹, 𝑇, 𝑅⟩ → (2nd ‘(1st𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
1817adantl 483 . . . . . 6 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩) → (2nd ‘(1st𝑏)) = (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)))
19 ot2ndg 7941 . . . . . . 7 ((𝐹𝑊𝑇𝑋𝑅𝑌) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
2019adantl 483 . . . . . 6 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (2nd ‘(1st ‘⟨𝐹, 𝑇, 𝑅⟩)) = 𝑇)
2118, 20sylan9eqr 2799 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (2nd ‘(1st𝑏)) = 𝑇)
223fveq2d 6851 . . . . 5 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (♯‘𝑠) = (♯‘𝑆))
2321, 22opeq12d 4843 . . . 4 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → ⟨(2nd ‘(1st𝑏)), (♯‘𝑠)⟩ = ⟨𝑇, (♯‘𝑆)⟩)
243, 23oveq12d 7380 . . 3 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (𝑠 substr ⟨(2nd ‘(1st𝑏)), (♯‘𝑠)⟩) = (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩))
2516, 24oveq12d 7380 . 2 (((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝑇, 𝑅⟩)) → (((𝑠 prefix (1st ‘(1st𝑏))) ++ (2nd𝑏)) ++ (𝑠 substr ⟨(2nd ‘(1st𝑏)), (♯‘𝑠)⟩)) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
26 elex 3466 . . 3 (𝑆𝑉𝑆 ∈ V)
2726adantr 482 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → 𝑆 ∈ V)
28 otex 5427 . . 3 𝐹, 𝑇, 𝑅⟩ ∈ V
2928a1i 11 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → ⟨𝐹, 𝑇, 𝑅⟩ ∈ V)
30 ovexd 7397 . 2 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)) ∈ V)
312, 25, 27, 29, 30ovmpod 7512 1 ((𝑆𝑉 ∧ (𝐹𝑊𝑇𝑋𝑅𝑌)) → (𝑆 splice ⟨𝐹, 𝑇, 𝑅⟩) = (((𝑆 prefix 𝐹) ++ 𝑅) ++ (𝑆 substr ⟨𝑇, (♯‘𝑆)⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3448  cop 4597  cotp 4599  cfv 6501  (class class class)co 7362  cmpo 7364  1st c1st 7924  2nd c2nd 7925  chash 14237   ++ cconcat 14465   substr csubstr 14535   prefix cpfx 14565   splice csplice 14644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-ot 4600  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6453  df-fun 6503  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-splice 14645
This theorem is referenced by:  splid  14648  spllen  14649  splfv1  14650  splfv2a  14651  splval2  14652  gsumspl  18661  efgredleme  19532  efgredlemc  19534  efgcpbllemb  19544  frgpuplem  19561  splfv3  31854  cycpmco2f1  32015  cycpmco2rn  32016  cycpmco2lem2  32018  cycpmco2lem3  32019  cycpmco2lem4  32020  cycpmco2lem5  32021  cycpmco2lem6  32022  cycpmco2  32024
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