Theorem wlkdlem4 30151

Description: Lemma 4 for wlkd 30152. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 23-Jan-2021.)

Hypotheses

Ref	Expression
wlkd.p	⊢ (𝜑 → 𝑃 ∈ Word V)
wlkd.f	⊢ (𝜑 → 𝐹 ∈ Word V)
wlkd.l	⊢ (𝜑 → (♯‘𝑃) = ((♯‘𝐹) + 1))
wlkd.e	⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
wlkd.n	⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))

Assertion

Ref	Expression
wlkdlem4	⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))

Distinct variable groups:   𝑘,𝐹   𝑃,𝑘   𝑘,𝐼   𝜑,𝑘

Proof of Theorem wlkdlem4

Step	Hyp	Ref	Expression
1		wlkd.e	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
2		wlkd.n	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
3		r19.26 3124	. . 3 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)) ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) ↔ (∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))))
4		df-ne 2958	. . . . . 6 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) ↔ ¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)))
5		ifpfal 1092	. . . . . 6 ⊢ (¬ (𝑃‘𝑘) = (𝑃‘(𝑘 + 1)) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
6	4, 5	sylbi 220	. . . . 5 ⊢ ((𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)) → (if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ↔ {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
7	6	biimparc 485	. . . 4 ⊢ (({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)) ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) → if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
8	7	ralimi 3101	. . 3 ⊢ (∀𝑘 ∈ (0..^(♯‘𝐹))({(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)) ∧ (𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
9	3, 8	sylbir 238	. 2 ⊢ ((∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1))) → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))
10	1, 2, 9	syl2anc 596	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  if-wif 1078   = wceq 1570   ∈ wcel 2145   ≠ wne 2957  ∀wral 3078  Vcvv 3453   ⊆ wss 3902  {csn 4587  {cpr 4589  ‘cfv 6537  (class class class)co 7417  0cc0 11128  1c1 11129   + caddc 11131  ..^cfzo 13713  ♯chash 14398  Word cword 14582

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-ne 2958  df-ral 3079

This theorem is used by:  wlkd  30152