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Theorem wlkdlem4 27394
Description: Lemma 4 for wlkd 27395. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 23-Jan-2021.)
Hypotheses
Ref Expression
wlkd.p (𝜑𝑃 ∈ Word V)
wlkd.f (𝜑𝐹 ∈ Word V)
wlkd.l (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))
wlkd.e (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
wlkd.n (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
Assertion
Ref Expression
wlkdlem4 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
Distinct variable groups:   𝑘,𝐹   𝑃,𝑘   𝑘,𝐼   𝜑,𝑘

Proof of Theorem wlkdlem4
StepHypRef Expression
1 wlkd.e . 2 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
2 wlkd.n . 2 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
3 r19.26 3167 . . 3 (∀𝑘 ∈ (0..^(♯‘𝐹))({(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)) ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))) ↔ (∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))))
4 df-ne 3014 . . . . . 6 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)))
5 ifpfal 1066 . . . . . 6 (¬ (𝑃𝑘) = (𝑃‘(𝑘 + 1)) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
64, 5sylbi 218 . . . . 5 ((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) → (if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
76biimparc 480 . . . 4 (({(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)) ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))) → if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
87ralimi 3157 . . 3 (∀𝑘 ∈ (0..^(♯‘𝐹))({(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)) ∧ (𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
93, 8sylbir 236 . 2 ((∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
101, 2, 9syl2anc 584 1 (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  if-wif 1054   = wceq 1528  wcel 2105  wne 3013  wral 3135  Vcvv 3492  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528  ..^cfzo 13021  chash 13678  Word cword 13849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-ifp 1055  df-ne 3014  df-ral 3140
This theorem is referenced by:  wlkd  27395
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