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

Theorem swrdval 13704
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 13702 . . 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 6434 . . . . 5 (𝑏 = ⟨𝐹, 𝐿⟩ → (1st𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
54adantl 475 . . . 4 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩) → (1st𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6 op1stg 7441 . . . . 5 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
763adant1 1166 . . . 4 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
85, 7sylan9eqr 2884 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → (1st𝑏) = 𝐹)
9 fveq2 6434 . . . . 5 (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
109adantl 475 . . . 4 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11 op2ndg 7442 . . . . 5 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12113adant1 1166 . . . 4 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
1310, 12sylan9eqr 2884 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd𝑏) = 𝐿)
14 simp2 1173 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (1st𝑏) = 𝐹)
15 simp3 1174 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (2nd𝑏) = 𝐿)
1614, 15oveq12d 6924 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → ((1st𝑏)..^(2nd𝑏)) = (𝐹..^𝐿))
17 simp1 1172 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → 𝑠 = 𝑆)
1817dmeqd 5559 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
1916, 18sseq12d 3860 . . . 4 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
2015, 14oveq12d 6924 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → ((2nd𝑏) − (1st𝑏)) = (𝐿𝐹))
2120oveq2d 6922 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (0..^((2nd𝑏) − (1st𝑏))) = (0..^(𝐿𝐹)))
2214oveq2d 6922 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑥 + (1st𝑏)) = (𝑥 + 𝐹))
2317, 22fveq12d 6441 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st𝑏))) = (𝑆‘(𝑥 + 𝐹)))
2421, 23mpteq12dv 4957 . . . 4 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))) = (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))))
2519, 24ifbieq1d 4330 . . 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 1496 . 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 3430 . . 3 (𝑆𝑉𝑆 ∈ V)
28273ad2ant1 1169 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29 opelxpi 5380 . . 3 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30293adant1 1166 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31 ovex 6938 . . . . 5 (0..^(𝐿𝐹)) ∈ V
3231mptex 6743 . . . 4 (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33 0ex 5015 . . . 4 ∅ ∈ V
3432, 33ifex 4355 . . 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 7049 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 386  w3a 1113   = wceq 1658  wcel 2166  Vcvv 3415  wss 3799  c0 4145  ifcif 4307  cop 4404  cmpt 4953   × cxp 5341  dom cdm 5343  cfv 6124  (class class class)co 6906  cmpt2 6908  1st c1st 7427  2nd c2nd 7428  0cc0 10253   + caddc 10256  cmin 10586  cz 11705  ..^cfzo 12761   substr csubstr 13701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-substr 13702
This theorem is referenced by:  swrd00  13705  swrdcl  13706  swrdval2  13707  swrdlend  13719  swrdnd  13720  swrdnd2  13721  swrd0  13724  repswswrd  13901
  Copyright terms: Public domain W3C validator