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Theorem crctcsh1wlkn0lem4 41011
Description: Lemma for crctcsh1wlkn0 41019. (Contributed by AV, 12-Mar-2021.)
Hypotheses
Ref Expression
crctcsh1wlkn0lem.s (𝜑𝑆 ∈ (1..^𝑁))
crctcsh1wlkn0lem.q 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
crctcsh1wlkn0lem.h 𝐻 = (𝐹 cyclShift 𝑆)
crctcsh1wlkn0lem.n 𝑁 = (#‘𝐹)
crctcsh1wlkn0lem.f (𝜑𝐹 ∈ Word 𝐴)
crctcsh1wlkn0lem.p (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
Assertion
Ref Expression
crctcsh1wlkn0lem4 (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))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
Allowed substitution hints:   𝐴(𝑥,𝑖,𝑗)   𝑃(𝑗)   𝑄(𝑥,𝑖,𝑗)   𝑆(𝑗)   𝐹(𝑥,𝑗)   𝐻(𝑥,𝑖,𝑗)   𝐼(𝑥,𝑗)   𝑁(𝑗)

Proof of Theorem crctcsh1wlkn0lem4
StepHypRef Expression
1 crctcsh1wlkn0lem.p . . . . 5 (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
2 crctcsh1wlkn0lem.s . . . . . . 7 (𝜑𝑆 ∈ (1..^𝑁))
3 elfzoelz 12294 . . . . . . . . . . 11 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℤ)
43zcnd 11315 . . . . . . . . . 10 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℂ)
54adantl 480 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ ℂ)
6 elfzoelz 12294 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℤ)
76zcnd 11315 . . . . . . . . . 10 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℂ)
87adantr 479 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℂ)
9 1cnd 9912 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 1 ∈ ℂ)
105, 8, 9add32d 10114 . . . . . . . 8 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
11 elfzo1 12340 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
12 nnnn0 11146 . . . . . . . . . . . . . . 15 (𝑆 ∈ ℕ → 𝑆 ∈ ℕ0)
13 elfzonn0 12335 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℕ0)
14 nn0addcl 11175 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑆 ∈ ℕ0) → (𝑗 + 𝑆) ∈ ℕ0)
1514ex 448 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ0 → (𝑆 ∈ ℕ0 → (𝑗 + 𝑆) ∈ ℕ0))
1613, 15syl 17 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑆 ∈ ℕ0 → (𝑗 + 𝑆) ∈ ℕ0))
1712, 16syl5com 31 . . . . . . . . . . . . . 14 (𝑆 ∈ ℕ → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
18173ad2ant1 1074 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
1911, 18sylbi 205 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
2019imp 443 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
21 fzo0ss1 12322 . . . . . . . . . . . . . 14 (1..^𝑁) ⊆ (0..^𝑁)
2221sseli 3563 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ (0..^𝑁))
23 elfzo0 12331 . . . . . . . . . . . . . 14 (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
2423simp2bi 1069 . . . . . . . . . . . . 13 (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
2522, 24syl 17 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ ℕ)
2625adantr 479 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑁 ∈ ℕ)
27 elfzo0 12331 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^(𝑁𝑆)) ↔ (𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)))
28 nn0re 11148 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℕ0𝑗 ∈ ℝ)
29 nnre 10874 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
30 nnre 10874 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
3129, 30anim12i 587 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
32313adant3 1073 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3311, 32sylbi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (1..^𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3428, 33anim12i 587 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
35 3anass 1034 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) ↔ (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
3634, 35sylibr 222 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
37 ltaddsub 10351 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑗 + 𝑆) < 𝑁𝑗 < (𝑁𝑆)))
3837bicomd 211 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑗 < (𝑁𝑆) ↔ (𝑗 + 𝑆) < 𝑁))
3936, 38syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 < (𝑁𝑆) ↔ (𝑗 + 𝑆) < 𝑁))
4039biimpd 217 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁))
4140ex 448 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ0 → (𝑆 ∈ (1..^𝑁) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁)))
4241com23 83 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → (𝑗 < (𝑁𝑆) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁)))
4342a1d 25 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ0 → ((𝑁𝑆) ∈ ℕ → (𝑗 < (𝑁𝑆) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))))
44433imp 1248 . . . . . . . . . . . . 13 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4527, 44sylbi 205 . . . . . . . . . . . 12 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4645impcom 444 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) < 𝑁)
47 elfzo0 12331 . . . . . . . . . . 11 ((𝑗 + 𝑆) ∈ (0..^𝑁) ↔ ((𝑗 + 𝑆) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑗 + 𝑆) < 𝑁))
4820, 26, 46, 47syl3anbrc 1238 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0..^𝑁))
4948adantr 479 . . . . . . . . 9 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑗 + 𝑆) ∈ (0..^𝑁))
50 fveq2 6088 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
5150adantl 480 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
52 oveq1 6534 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 𝑆) → (𝑖 + 1) = ((𝑗 + 𝑆) + 1))
5352fveq2d 6092 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃‘(𝑖 + 1)) = (𝑃‘((𝑗 + 𝑆) + 1)))
54 simpr 475 . . . . . . . . . . . . 13 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
5554fveq2d 6092 . . . . . . . . . . . 12 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑃‘((𝑗 + 𝑆) + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
5653, 55sylan9eqr 2665 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝑃‘(𝑖 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
5751, 56eqeq12d 2624 . . . . . . . . . 10 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → ((𝑃𝑖) = (𝑃‘(𝑖 + 1)) ↔ (𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆))))
58 fveq2 6088 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 𝑆) → (𝐹𝑖) = (𝐹‘(𝑗 + 𝑆)))
5958fveq2d 6092 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
6050sneqd 4136 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → {(𝑃𝑖)} = {(𝑃‘(𝑗 + 𝑆))})
6159, 60eqeq12d 2624 . . . . . . . . . . 11 (𝑖 = (𝑗 + 𝑆) → ((𝐼‘(𝐹𝑖)) = {(𝑃𝑖)} ↔ (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}))
6261adantl 480 . . . . . . . . . 10 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → ((𝐼‘(𝐹𝑖)) = {(𝑃𝑖)} ↔ (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}))
6351, 56preq12d 4219 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → {(𝑃𝑖), (𝑃‘(𝑖 + 1))} = {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))})
6459adantl 480 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
6563, 64sseq12d 3596 . . . . . . . . . 10 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → ({(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖)) ↔ {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆)))))
6657, 62, 65ifpbi123d 1020 . . . . . . . . 9 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) ↔ if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
6749, 66rspcdv 3284 . . . . . . . 8 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
6810, 67mpdan 698 . . . . . . 7 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
692, 68sylan 486 . . . . . 6 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
7069ex 448 . . . . 5 (𝜑 → (𝑗 ∈ (0..^(𝑁𝑆)) → (∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))) → if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆)))))))
711, 70mpid 42 . . . 4 (𝜑 → (𝑗 ∈ (0..^(𝑁𝑆)) → if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
7271imp 443 . . 3 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆)))))
73 elfzofz 12309 . . . . 5 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ (0...(𝑁𝑆)))
74 crctcsh1wlkn0lem.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
752, 74crctcsh1wlkn0lem2 41009 . . . . 5 ((𝜑𝑗 ∈ (0...(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
7673, 75sylan2 489 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
77 fzofzp1 12386 . . . . 5 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 1) ∈ (0...(𝑁𝑆)))
782, 74crctcsh1wlkn0lem2 41009 . . . . 5 ((𝜑 ∧ (𝑗 + 1) ∈ (0...(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
7977, 78sylan2 489 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
80 crctcsh1wlkn0lem.h . . . . . . 7 𝐻 = (𝐹 cyclShift 𝑆)
8180fveq1i 6089 . . . . . 6 (𝐻𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗)
82 crctcsh1wlkn0lem.f . . . . . . . . 9 (𝜑𝐹 ∈ Word 𝐴)
8382adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝐹 ∈ Word 𝐴)
842, 6syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ ℤ)
8584adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℤ)
86 nnz 11232 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
8786adantl 480 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
88 nnz 11232 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
8988adantr 479 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ ℤ)
9087, 89zsubcld 11319 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
9112nn0ge0d 11201 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 0 ≤ 𝑆)
9291adantr 479 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑆)
93 subge02 10393 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9430, 29, 93syl2anr 493 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9592, 94mpbid 220 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ≤ 𝑁)
9690, 87, 953jca 1234 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
97963adant3 1073 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
9811, 97sylbi 205 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
99 eluz2 11525 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘(𝑁𝑆)) ↔ ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
10098, 99sylibr 222 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘(𝑁𝑆)))
101 fzoss2 12320 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘(𝑁𝑆)) → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
1022, 100, 1013syl 18 . . . . . . . . . 10 (𝜑 → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
103102sselda 3567 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^𝑁))
104 crctcsh1wlkn0lem.n . . . . . . . . . 10 𝑁 = (#‘𝐹)
105104oveq2i 6538 . . . . . . . . 9 (0..^𝑁) = (0..^(#‘𝐹))
106103, 105syl6eleq 2697 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^(#‘𝐹)))
107 cshwidxmod 13346 . . . . . . . 8 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
10883, 85, 106, 107syl3anc 1317 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
109104eqcomi 2618 . . . . . . . . . 10 (#‘𝐹) = 𝑁
110109oveq2i 6538 . . . . . . . . 9 ((𝑗 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod 𝑁)
11118imp 443 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
112 nnm1nn0 11181 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1131123ad2ant2 1075 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 − 1) ∈ ℕ0)
114113adantr 479 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑁 − 1) ∈ ℕ0)
11528, 32anim12i 587 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
116115, 35sylibr 222 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
117116, 38syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 < (𝑁𝑆) ↔ (𝑗 + 𝑆) < 𝑁))
118123ad2ant1 1074 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℕ0)
119118, 14sylan2 489 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℕ0)
120119nn0zd 11312 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℤ)
121863ad2ant2 1075 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℤ)
122121adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → 𝑁 ∈ ℤ)
123 zltlem1 11263 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
124120, 122, 123syl2anc 690 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
125124biimpd 217 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
126117, 125sylbid 228 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
127126impancom 454 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
1281273adant2 1072 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
12927, 128sylbi 205 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
130129impcom 444 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ≤ (𝑁 − 1))
131111, 114, 1303jca 1234 . . . . . . . . . . . . 13 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
13211, 131sylanb 487 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
133 elfz2nn0 12255 . . . . . . . . . . . 12 ((𝑗 + 𝑆) ∈ (0...(𝑁 − 1)) ↔ ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
134132, 133sylibr 222 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0...(𝑁 − 1)))
135 zaddcl 11250 . . . . . . . . . . . . 13 ((𝑗 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑗 + 𝑆) ∈ ℤ)
1363, 6, 135syl2anr 493 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℤ)
137 zmodid2 12515 . . . . . . . . . . . 12 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
138136, 26, 137syl2anc 690 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
139134, 138mpbird 245 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
1402, 139sylan 486 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
141110, 140syl5eq 2655 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod (#‘𝐹)) = (𝑗 + 𝑆))
142141fveq2d 6092 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) = (𝐹‘(𝑗 + 𝑆)))
143108, 142eqtrd 2643 . . . . . 6 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘(𝑗 + 𝑆)))
14481, 143syl5eq 2655 . . . . 5 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐻𝑗) = (𝐹‘(𝑗 + 𝑆)))
145144fveq2d 6092 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
146 simp1 1053 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
147 simp2 1054 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
148146, 147eqeq12d 2624 . . . . 5 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → ((𝑄𝑗) = (𝑄‘(𝑗 + 1)) ↔ (𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆))))
149 simp3 1055 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
150146sneqd 4136 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → {(𝑄𝑗)} = {(𝑃‘(𝑗 + 𝑆))})
151149, 150eqeq12d 2624 . . . . 5 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → ((𝐼‘(𝐻𝑗)) = {(𝑄𝑗)} ↔ (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}))
152146, 147preq12d 4219 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → {(𝑄𝑗), (𝑄‘(𝑗 + 1))} = {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))})
153152, 149sseq12d 3596 . . . . 5 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → ({(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)) ↔ {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆)))))
154148, 151, 153ifpbi123d 1020 . . . 4 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
15576, 79, 145, 154syl3anc 1317 . . 3 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ if-((𝑃‘(𝑗 + 𝑆)) = (𝑃‘((𝑗 + 1) + 𝑆)), (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}, {(𝑃‘(𝑗 + 𝑆)), (𝑃‘((𝑗 + 1) + 𝑆))} ⊆ (𝐼‘(𝐹‘(𝑗 + 𝑆))))))
15672, 155mpbird 245 . 2 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
157156ralrimiva 2948 1 (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  if-wif 1005  w3a 1030   = wceq 1474  wcel 1976  wral 2895  wss 3539  ifcif 4035  {csn 4124  {cpr 4126   class class class wbr 4577  cmpt 4637  cfv 5790  (class class class)co 6527  cc 9790  cr 9791  0cc0 9792  1c1 9793   + caddc 9795   < clt 9930  cle 9931  cmin 10117  cn 10867  0cn0 11139  cz 11210  cuz 11519  ...cfz 12152  ..^cfzo 12289   mod cmo 12485  #chash 12934  Word cword 13092   cyclShift ccsh 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-oadd 7428  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-n0 11140  df-z 11211  df-uz 11520  df-rp 11665  df-fz 12153  df-fzo 12290  df-fl 12410  df-mod 12486  df-hash 12935  df-word 13100  df-concat 13102  df-substr 13104  df-csh 13332
This theorem is referenced by:  crctcsh1wlkn0lem7  41014
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