MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  swrdval Structured version   Visualization version   GIF version

Theorem swrdval 13126
Description: Value of a subword. (Contributed by Stefan O'Rear, 15-Aug-2015.)
Assertion
Ref Expression
swrdval ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
Distinct variable groups:   𝑥,𝑆   𝑥,𝐹   𝑥,𝐿   𝑥,𝑉

Proof of Theorem swrdval
Dummy variables 𝑠 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-substr 13015 . . 3 substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅))
21a1i 11 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → substr = (𝑠 ∈ V, 𝑏 ∈ (ℤ × ℤ) ↦ if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅)))
3 simprl 789 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → 𝑠 = 𝑆)
4 fveq2 5986 . . . . 5 (𝑏 = ⟨𝐹, 𝐿⟩ → (1st𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
54adantl 480 . . . 4 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩) → (1st𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6 op1stg 6945 . . . . 5 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
763adant1 1071 . . . 4 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
85, 7sylan9eqr 2570 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → (1st𝑏) = 𝐹)
9 fveq2 5986 . . . . 5 (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
109adantl 480 . . . 4 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11 op2ndg 6946 . . . . 5 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12113adant1 1071 . . . 4 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
1310, 12sylan9eqr 2570 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd𝑏) = 𝐿)
14 simp2 1054 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (1st𝑏) = 𝐹)
15 simp3 1055 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (2nd𝑏) = 𝐿)
1614, 15oveq12d 6443 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → ((1st𝑏)..^(2nd𝑏)) = (𝐹..^𝐿))
17 simp1 1053 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → 𝑠 = 𝑆)
1817dmeqd 5139 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
1916, 18sseq12d 3501 . . . 4 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
2015, 14oveq12d 6443 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → ((2nd𝑏) − (1st𝑏)) = (𝐿𝐹))
2120oveq2d 6441 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (0..^((2nd𝑏) − (1st𝑏))) = (0..^(𝐿𝐹)))
2214oveq2d 6441 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑥 + (1st𝑏)) = (𝑥 + 𝐹))
2317, 22fveq12d 5992 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st𝑏))) = (𝑆‘(𝑥 + 𝐹)))
2421, 23mpteq12dv 4561 . . . 4 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))) = (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))))
2519, 24ifbieq1d 3962 . . 3 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
263, 8, 13, 25syl3anc 1317 . 2 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → if(((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠, (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))), ∅) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
27 elex 3089 . . 3 (𝑆𝑉𝑆 ∈ V)
28273ad2ant1 1074 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29 opelxpi 4966 . . 3 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30293adant1 1071 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31 ovex 6453 . . . . 5 (0..^(𝐿𝐹)) ∈ V
3231mptex 6266 . . . 4 (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33 0ex 4617 . . . 4 ∅ ∈ V
3432, 33ifex 4009 . . 3 if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ V
3534a1i 11 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅) ∈ V)
362, 26, 28, 30, 35ovmpt2d 6562 1 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (𝑆 substr ⟨𝐹, 𝐿⟩) = if((𝐹..^𝐿) ⊆ dom 𝑆, (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))), ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1938  Vcvv 3077  wss 3444  c0 3777  ifcif 3939  cop 4034  cmpt 4541   × cxp 4930  dom cdm 4932  cfv 5689  (class class class)co 6425  cmpt2 6427  1st c1st 6931  2nd c2nd 6932  0cc0 9690   + caddc 9693  cmin 10016  cz 11117  ..^cfzo 12201   substr csubstr 13007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6722
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-1st 6933  df-2nd 6934  df-substr 13015
This theorem is referenced by:  swrd00  13127  swrdcl  13128  swrdval2  13129  swrdlend  13140  swrdnd  13141  swrdnd2  13142  swrd0  13143  repswswrd  13239
  Copyright terms: Public domain W3C validator