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

Theorem swrdval 14589
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 14587 . . 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 768 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → 𝑠 = 𝑆)
4 fveq2 6881 . . . . 5 (𝑏 = ⟨𝐹, 𝐿⟩ → (1st𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
54adantl 481 . . . 4 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩) → (1st𝑏) = (1st ‘⟨𝐹, 𝐿⟩))
6 op1stg 7980 . . . . 5 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
763adant1 1127 . . . 4 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (1st ‘⟨𝐹, 𝐿⟩) = 𝐹)
85, 7sylan9eqr 2786 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → (1st𝑏) = 𝐹)
9 fveq2 6881 . . . . 5 (𝑏 = ⟨𝐹, 𝐿⟩ → (2nd𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
109adantl 481 . . . 4 ((𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩) → (2nd𝑏) = (2nd ‘⟨𝐹, 𝐿⟩))
11 op2ndg 7981 . . . . 5 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
12113adant1 1127 . . . 4 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → (2nd ‘⟨𝐹, 𝐿⟩) = 𝐿)
1310, 12sylan9eqr 2786 . . 3 (((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) ∧ (𝑠 = 𝑆𝑏 = ⟨𝐹, 𝐿⟩)) → (2nd𝑏) = 𝐿)
14 simp2 1134 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (1st𝑏) = 𝐹)
15 simp3 1135 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (2nd𝑏) = 𝐿)
1614, 15oveq12d 7419 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → ((1st𝑏)..^(2nd𝑏)) = (𝐹..^𝐿))
17 simp1 1133 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → 𝑠 = 𝑆)
1817dmeqd 5895 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → dom 𝑠 = dom 𝑆)
1916, 18sseq12d 4007 . . . 4 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (((1st𝑏)..^(2nd𝑏)) ⊆ dom 𝑠 ↔ (𝐹..^𝐿) ⊆ dom 𝑆))
2015, 14oveq12d 7419 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → ((2nd𝑏) − (1st𝑏)) = (𝐿𝐹))
2120oveq2d 7417 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (0..^((2nd𝑏) − (1st𝑏))) = (0..^(𝐿𝐹)))
2214oveq2d 7417 . . . . . 6 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑥 + (1st𝑏)) = (𝑥 + 𝐹))
2317, 22fveq12d 6888 . . . . 5 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑠‘(𝑥 + (1st𝑏))) = (𝑆‘(𝑥 + 𝐹)))
2421, 23mpteq12dv 5229 . . . 4 ((𝑠 = 𝑆 ∧ (1st𝑏) = 𝐹 ∧ (2nd𝑏) = 𝐿) → (𝑥 ∈ (0..^((2nd𝑏) − (1st𝑏))) ↦ (𝑠‘(𝑥 + (1st𝑏)))) = (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))))
2519, 24ifbieq1d 4544 . . 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 1368 . 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 3485 . . 3 (𝑆𝑉𝑆 ∈ V)
28273ad2ant1 1130 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → 𝑆 ∈ V)
29 opelxpi 5703 . . 3 ((𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
30293adant1 1127 . 2 ((𝑆𝑉𝐹 ∈ ℤ ∧ 𝐿 ∈ ℤ) → ⟨𝐹, 𝐿⟩ ∈ (ℤ × ℤ))
31 ovex 7434 . . . . 5 (0..^(𝐿𝐹)) ∈ V
3231mptex 7216 . . . 4 (𝑥 ∈ (0..^(𝐿𝐹)) ↦ (𝑆‘(𝑥 + 𝐹))) ∈ V
33 0ex 5297 . . . 4 ∅ ∈ V
3432, 33ifex 4570 . . 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, 35ovmpod 7552 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 395  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3466  wss 3940  c0 4314  ifcif 4520  cop 4626  cmpt 5221   × cxp 5664  dom cdm 5666  cfv 6533  (class class class)co 7401  cmpo 7403  1st c1st 7966  2nd c2nd 7967  0cc0 11105   + caddc 11108  cmin 11440  cz 12554  ..^cfzo 13623   substr csubstr 14586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-substr 14587
This theorem is referenced by:  swrd00  14590  swrdcl  14591  swrdval2  14592  swrdlend  14599  swrdnd  14600  swrdnd2  14601  swrd0  14604  repswswrd  14730
  Copyright terms: Public domain W3C validator