Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crctcsh1wlkn0lem4 Structured version   Visualization version   GIF version

Theorem crctcsh1wlkn0lem4 41016
 Description: Lemma for crctcsh1wlkn0 41024. (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 12339 . . . . . . . . . . 11 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℤ)
43zcnd 11359 . . . . . . . . . 10 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℂ)
54adantl 481 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ ℂ)
6 elfzoelz 12339 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℤ)
76zcnd 11359 . . . . . . . . . 10 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℂ)
87adantr 480 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℂ)
9 1cnd 9935 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 1 ∈ ℂ)
105, 8, 9add32d 10142 . . . . . . . 8 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
11 elfzo1 12385 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
12 nnnn0 11176 . . . . . . . . . . . . . . 15 (𝑆 ∈ ℕ → 𝑆 ∈ ℕ0)
13 elfzonn0 12380 . . . . . . . . . . . . . . . 16 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℕ0)
14 nn0addcl 11205 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑆 ∈ ℕ0) → (𝑗 + 𝑆) ∈ ℕ0)
1514ex 449 . . . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
1911, 18sylbi 206 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
2019imp 444 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
21 fzo0ss1 12367 . . . . . . . . . . . . . 14 (1..^𝑁) ⊆ (0..^𝑁)
2221sseli 3564 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ (0..^𝑁))
23 elfzo0 12376 . . . . . . . . . . . . . 14 (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
2423simp2bi 1070 . . . . . . . . . . . . 13 (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
2522, 24syl 17 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ ℕ)
2625adantr 480 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑁 ∈ ℕ)
27 elfzo0 12376 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^(𝑁𝑆)) ↔ (𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)))
28 nn0re 11178 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℕ0𝑗 ∈ ℝ)
29 nnre 10904 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
30 nnre 10904 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
3129, 30anim12i 588 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
32313adant3 1074 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3311, 32sylbi 206 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (1..^𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3428, 33anim12i 588 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
35 3anass 1035 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) ↔ (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
3634, 35sylibr 223 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
37 ltaddsub 10381 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑗 + 𝑆) < 𝑁𝑗 < (𝑁𝑆)))
3837bicomd 212 . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁))
4140ex 449 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ0 → (𝑆 ∈ (1..^𝑁) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁)))
4241com23 84 . . . . . . . . . . . . . . 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 1249 . . . . . . . . . . . . 13 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4527, 44sylbi 206 . . . . . . . . . . . 12 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4645impcom 445 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) < 𝑁)
47 elfzo0 12376 . . . . . . . . . . 11 ((𝑗 + 𝑆) ∈ (0..^𝑁) ↔ ((𝑗 + 𝑆) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑗 + 𝑆) < 𝑁))
4820, 26, 46, 47syl3anbrc 1239 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0..^𝑁))
4948adantr 480 . . . . . . . . 9 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑗 + 𝑆) ∈ (0..^𝑁))
50 fveq2 6103 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
5150adantl 481 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
52 oveq1 6556 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 𝑆) → (𝑖 + 1) = ((𝑗 + 𝑆) + 1))
5352fveq2d 6107 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃‘(𝑖 + 1)) = (𝑃‘((𝑗 + 𝑆) + 1)))
54 simpr 476 . . . . . . . . . . . . 13 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
5554fveq2d 6107 . . . . . . . . . . . 12 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑃‘((𝑗 + 𝑆) + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
5653, 55sylan9eqr 2666 . . . . . . . . . . 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 2625 . . . . . . . . . 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 6103 . . . . . . . . . . . . 13 (𝑖 = (𝑗 + 𝑆) → (𝐹𝑖) = (𝐹‘(𝑗 + 𝑆)))
5958fveq2d 6107 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
6050sneqd 4137 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → {(𝑃𝑖)} = {(𝑃‘(𝑗 + 𝑆))})
6159, 60eqeq12d 2625 . . . . . . . . . . 11 (𝑖 = (𝑗 + 𝑆) → ((𝐼‘(𝐹𝑖)) = {(𝑃𝑖)} ↔ (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}))
6261adantl 481 . . . . . . . . . 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 4220 . . . . . . . . . . 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 481 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
6563, 64sseq12d 3597 . . . . . . . . . 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 1021 . . . . . . . . 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 3285 . . . . . . . 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 699 . . . . . . 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 487 . . . . . 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 449 . . . . 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 43 . . . 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 444 . . 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 12354 . . . . 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 41014 . . . . 5 ((𝜑𝑗 ∈ (0...(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
7673, 75sylan2 490 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
77 fzofzp1 12431 . . . . 5 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 1) ∈ (0...(𝑁𝑆)))
782, 74crctcsh1wlkn0lem2 41014 . . . . 5 ((𝜑 ∧ (𝑗 + 1) ∈ (0...(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
7977, 78sylan2 490 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
80 crctcsh1wlkn0lem.h . . . . . . 7 𝐻 = (𝐹 cyclShift 𝑆)
8180fveq1i 6104 . . . . . 6 (𝐻𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗)
82 crctcsh1wlkn0lem.f . . . . . . . . 9 (𝜑𝐹 ∈ Word 𝐴)
8382adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝐹 ∈ Word 𝐴)
842, 6syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ ℤ)
8584adantr 480 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℤ)
86 nnz 11276 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
8786adantl 481 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
88 nnz 11276 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
8988adantr 480 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ ℤ)
9087, 89zsubcld 11363 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
9112nn0ge0d 11231 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 0 ≤ 𝑆)
9291adantr 480 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑆)
93 subge02 10423 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9430, 29, 93syl2anr 494 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9592, 94mpbid 221 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ≤ 𝑁)
9690, 87, 953jca 1235 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
97963adant3 1074 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
9811, 97sylbi 206 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
99 eluz2 11569 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘(𝑁𝑆)) ↔ ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
10098, 99sylibr 223 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘(𝑁𝑆)))
101 fzoss2 12365 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘(𝑁𝑆)) → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
1022, 100, 1013syl 18 . . . . . . . . . 10 (𝜑 → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
103102sselda 3568 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^𝑁))
104 crctcsh1wlkn0lem.n . . . . . . . . . 10 𝑁 = (#‘𝐹)
105104oveq2i 6560 . . . . . . . . 9 (0..^𝑁) = (0..^(#‘𝐹))
106103, 105syl6eleq 2698 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^(#‘𝐹)))
107 cshwidxmod 13400 . . . . . . . 8 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
10883, 85, 106, 107syl3anc 1318 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
109104eqcomi 2619 . . . . . . . . . 10 (#‘𝐹) = 𝑁
110109oveq2i 6560 . . . . . . . . 9 ((𝑗 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod 𝑁)
11118imp 444 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
112 nnm1nn0 11211 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1131123ad2ant2 1076 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 − 1) ∈ ℕ0)
114113adantr 480 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑁 − 1) ∈ ℕ0)
11528, 32anim12i 588 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
116115, 35sylibr 223 . . . . . . . . . . . . . . . . . . . 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 1075 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℕ0)
119118, 14sylan2 490 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℕ0)
120119nn0zd 11356 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℤ)
121863ad2ant2 1076 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℤ)
122121adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → 𝑁 ∈ ℤ)
123 zltlem1 11307 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
124120, 122, 123syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
125124biimpd 218 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
126117, 125sylbid 229 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
127126impancom 455 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
1281273adant2 1073 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
12927, 128sylbi 206 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
130129impcom 445 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ≤ (𝑁 − 1))
131111, 114, 1303jca 1235 . . . . . . . . . . . . 13 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
13211, 131sylanb 488 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
133 elfz2nn0 12300 . . . . . . . . . . . 12 ((𝑗 + 𝑆) ∈ (0...(𝑁 − 1)) ↔ ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
134132, 133sylibr 223 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0...(𝑁 − 1)))
135 zaddcl 11294 . . . . . . . . . . . . 13 ((𝑗 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑗 + 𝑆) ∈ ℤ)
1363, 6, 135syl2anr 494 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℤ)
137 zmodid2 12560 . . . . . . . . . . . 12 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
138136, 26, 137syl2anc 691 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
139134, 138mpbird 246 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
1402, 139sylan 487 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
141110, 140syl5eq 2656 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod (#‘𝐹)) = (𝑗 + 𝑆))
142141fveq2d 6107 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) = (𝐹‘(𝑗 + 𝑆)))
143108, 142eqtrd 2644 . . . . . 6 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘(𝑗 + 𝑆)))
14481, 143syl5eq 2656 . . . . 5 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐻𝑗) = (𝐹‘(𝑗 + 𝑆)))
145144fveq2d 6107 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
146 simp1 1054 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
147 simp2 1055 . . . . . 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 2625 . . . . 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 1056 . . . . . 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 4137 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → {(𝑄𝑗)} = {(𝑃‘(𝑗 + 𝑆))})
151149, 150eqeq12d 2625 . . . . 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 4220 . . . . . 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 3597 . . . . 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 1021 . . . 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 1318 . . 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 246 . 2 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
157156ralrimiva 2949 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 195   ∧ wa 383  if-wif 1006   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  ifcif 4036  {csn 4125  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℂcc 9813  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334   mod cmo 12530  #chash 12979  Word cword 13146   cyclShift ccsh 13385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386 This theorem is referenced by:  crctcsh1wlkn0lem7  41019
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