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Theorem crctcshwlkn0lem7 29836
Description: Lemma for crctcshwlkn0 29841. (Contributed by AV, 12-Mar-2021.)
Hypotheses
Ref Expression
crctcshwlkn0lem.s (𝜑𝑆 ∈ (1..^𝑁))
crctcshwlkn0lem.q 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
crctcshwlkn0lem.h 𝐻 = (𝐹 cyclShift 𝑆)
crctcshwlkn0lem.n 𝑁 = (♯‘𝐹)
crctcshwlkn0lem.f (𝜑𝐹 ∈ Word 𝐴)
crctcshwlkn0lem.p (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
crctcshwlkn0lem.e (𝜑 → (𝑃𝑁) = (𝑃‘0))
Assertion
Ref Expression
crctcshwlkn0lem7 (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
Distinct variable groups:   𝑥,𝑁   𝑥,𝑃   𝑥,𝑆   𝜑,𝑥   𝑖,𝐹   𝑖,𝐼   𝑖,𝑁   𝑃,𝑖   𝑆,𝑖   𝜑,𝑖,𝑗   𝑥,𝑗   𝑗,𝐼   𝑗,𝐻   𝑗,𝑁   𝑄,𝑗   𝑆,𝑗
Allowed substitution hints:   𝐴(𝑥,𝑖,𝑗)   𝑃(𝑗)   𝑄(𝑥,𝑖)   𝐹(𝑥,𝑗)   𝐻(𝑥,𝑖)   𝐼(𝑥)

Proof of Theorem crctcshwlkn0lem7
StepHypRef Expression
1 crctcshwlkn0lem.s . . . . . 6 (𝜑𝑆 ∈ (1..^𝑁))
2 crctcshwlkn0lem.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
3 crctcshwlkn0lem.h . . . . . 6 𝐻 = (𝐹 cyclShift 𝑆)
4 crctcshwlkn0lem.n . . . . . 6 𝑁 = (♯‘𝐹)
5 crctcshwlkn0lem.f . . . . . 6 (𝜑𝐹 ∈ Word 𝐴)
6 crctcshwlkn0lem.p . . . . . 6 (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
71, 2, 3, 4, 5, 6crctcshwlkn0lem4 29833 . . . . 5 (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
8 eqidd 2738 . . . . . . 7 (𝜑 → (𝑁𝑆) = (𝑁𝑆))
9 crctcshwlkn0lem.e . . . . . . . 8 (𝜑 → (𝑃𝑁) = (𝑃‘0))
101, 2, 3, 4, 5, 6, 9crctcshwlkn0lem6 29835 . . . . . . 7 ((𝜑 ∧ (𝑁𝑆) = (𝑁𝑆)) → if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
118, 10mpdan 687 . . . . . 6 (𝜑 → if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
12 ovex 7464 . . . . . . 7 (𝑁𝑆) ∈ V
13 wkslem1 29625 . . . . . . 7 (𝑗 = (𝑁𝑆) → (if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆))))))
1412, 13ralsn 4681 . . . . . 6 (∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
1511, 14sylibr 234 . . . . 5 (𝜑 → ∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
16 ralunb 4197 . . . . 5 (∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ (∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ∧ ∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
177, 15, 16sylanbrc 583 . . . 4 (𝜑 → ∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
18 elfzo1 13752 . . . . . 6 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
19 nnz 12634 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
20 nnz 12634 . . . . . . . . . . 11 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
21 zsubcl 12659 . . . . . . . . . . 11 ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
2219, 20, 21syl2anr 597 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
23223adant3 1133 . . . . . . . . 9 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℤ)
24 nnre 12273 . . . . . . . . . . 11 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
25 nnre 12273 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
26 posdif 11756 . . . . . . . . . . . 12 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 ↔ 0 < (𝑁𝑆)))
27 0re 11263 . . . . . . . . . . . . 13 0 ∈ ℝ
28 resubcl 11573 . . . . . . . . . . . . . 14 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
2928ancoms 458 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
30 ltle 11349 . . . . . . . . . . . . 13 ((0 ∈ ℝ ∧ (𝑁𝑆) ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3127, 29, 30sylancr 587 . . . . . . . . . . . 12 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3226, 31sylbid 240 . . . . . . . . . . 11 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
3324, 25, 32syl2an 596 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
34333impia 1118 . . . . . . . . 9 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 0 ≤ (𝑁𝑆))
35 elnn0z 12626 . . . . . . . . 9 ((𝑁𝑆) ∈ ℕ0 ↔ ((𝑁𝑆) ∈ ℤ ∧ 0 ≤ (𝑁𝑆)))
3623, 34, 35sylanbrc 583 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ0)
37 elnn0uz 12923 . . . . . . . 8 ((𝑁𝑆) ∈ ℕ0 ↔ (𝑁𝑆) ∈ (ℤ‘0))
3836, 37sylib 218 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ (ℤ‘0))
39 fzosplitsn 13814 . . . . . . 7 ((𝑁𝑆) ∈ (ℤ‘0) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4038, 39syl 17 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4118, 40sylbi 217 . . . . 5 (𝑆 ∈ (1..^𝑁) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
421, 41syl 17 . . . 4 (𝜑 → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4317, 42raleqtrrdv 3330 . . 3 (𝜑 → ∀𝑗 ∈ (0..^((𝑁𝑆) + 1))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
441, 2, 3, 4, 5, 6crctcshwlkn0lem5 29834 . . 3 (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
45 ralunb 4197 . . 3 (∀𝑗 ∈ ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ (∀𝑗 ∈ (0..^((𝑁𝑆) + 1))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ∧ ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
4643, 44, 45sylanbrc 583 . 2 (𝜑 → ∀𝑗 ∈ ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
47 nnsub 12310 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 ↔ (𝑁𝑆) ∈ ℕ))
4847biimp3a 1471 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ)
49 nnnn0 12533 . . . . . 6 ((𝑁𝑆) ∈ ℕ → (𝑁𝑆) ∈ ℕ0)
50 peano2nn0 12566 . . . . . 6 ((𝑁𝑆) ∈ ℕ0 → ((𝑁𝑆) + 1) ∈ ℕ0)
5148, 49, 503syl 18 . . . . 5 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ ℕ0)
52 nnnn0 12533 . . . . . 6 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
53523ad2ant2 1135 . . . . 5 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℕ0)
5425anim1i 615 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
5554ancoms 458 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
56 crctcshwlkn0lem1 29830 . . . . . . 7 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
5755, 56syl 17 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
58573adant3 1133 . . . . 5 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ≤ 𝑁)
59 elfz2nn0 13658 . . . . 5 (((𝑁𝑆) + 1) ∈ (0...𝑁) ↔ (((𝑁𝑆) + 1) ∈ ℕ0𝑁 ∈ ℕ0 ∧ ((𝑁𝑆) + 1) ≤ 𝑁))
6051, 53, 58, 59syl3anbrc 1344 . . . 4 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
6118, 60sylbi 217 . . 3 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
62 fzosplit 13732 . . 3 (((𝑁𝑆) + 1) ∈ (0...𝑁) → (0..^𝑁) = ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁)))
631, 61, 623syl 18 . 2 (𝜑 → (0..^𝑁) = ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁)))
6446, 63raleqtrrdv 3330 1 (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  if-wif 1063  w3a 1087   = wceq 1540  wcel 2108  wral 3061  cun 3949  wss 3951  ifcif 4525  {csn 4626  {cpr 4628   class class class wbr 5143  cmpt 5225  cfv 6561  (class class class)co 7431  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cle 11296  cmin 11492  cn 12266  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552   cyclShift ccsh 14826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-hash 14370  df-word 14553  df-concat 14609  df-substr 14679  df-pfx 14709  df-csh 14827
This theorem is referenced by:  crctcshwlkn0  29841
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