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Theorem crctcshwlkn0lem4 27505
Description: Lemma for crctcshwlkn0 27513. (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))))
Assertion
Ref Expression
crctcshwlkn0lem4 (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 crctcshwlkn0lem4
StepHypRef Expression
1 crctcshwlkn0lem.p . . . . 5 (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
2 crctcshwlkn0lem.s . . . . . . 7 (𝜑𝑆 ∈ (1..^𝑁))
3 elfzoelz 13028 . . . . . . . . . . 11 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℤ)
43zcnd 12077 . . . . . . . . . 10 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℂ)
54adantl 482 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ ℂ)
6 elfzoelz 13028 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℤ)
76zcnd 12077 . . . . . . . . . 10 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℂ)
87adantr 481 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℂ)
9 1cnd 10625 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 1 ∈ ℂ)
105, 8, 9add32d 10856 . . . . . . . 8 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
11 elfzo1 13077 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
12 elfzonn0 13072 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℕ0)
13 nnnn0 11893 . . . . . . . . . . . . . . 15 (𝑆 ∈ ℕ → 𝑆 ∈ ℕ0)
14 nn0addcl 11921 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ0𝑆 ∈ ℕ0) → (𝑗 + 𝑆) ∈ ℕ0)
1514ex 413 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → (𝑆 ∈ ℕ0 → (𝑗 + 𝑆) ∈ ℕ0))
1612, 13, 15syl2imc 41 . . . . . . . . . . . . . 14 (𝑆 ∈ ℕ → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
17163ad2ant1 1127 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
1811, 17sylbi 218 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
1918imp 407 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
20 fzo0ss1 13057 . . . . . . . . . . . . . 14 (1..^𝑁) ⊆ (0..^𝑁)
2120sseli 3967 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ (0..^𝑁))
22 elfzo0 13068 . . . . . . . . . . . . . 14 (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
2322simp2bi 1140 . . . . . . . . . . . . 13 (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
2421, 23syl 17 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ ℕ)
2524adantr 481 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑁 ∈ ℕ)
26 elfzo0 13068 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^(𝑁𝑆)) ↔ (𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)))
27 nn0re 11895 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℕ0𝑗 ∈ ℝ)
28 nnre 11634 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
29 nnre 11634 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
3028, 29anim12i 612 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
31303adant3 1126 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3211, 31sylbi 218 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (1..^𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3327, 32anim12i 612 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
34 3anass 1089 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) ↔ (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
3533, 34sylibr 235 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
36 ltaddsub 11103 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑗 + 𝑆) < 𝑁𝑗 < (𝑁𝑆)))
3736bicomd 224 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑗 < (𝑁𝑆) ↔ (𝑗 + 𝑆) < 𝑁))
3835, 37syl 17 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 < (𝑁𝑆) ↔ (𝑗 + 𝑆) < 𝑁))
3938biimpd 230 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁))
4039ex 413 . . . . . . . . . . . . . . . 16 (𝑗 ∈ ℕ0 → (𝑆 ∈ (1..^𝑁) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁)))
4140com23 86 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → (𝑗 < (𝑁𝑆) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁)))
4241a1d 25 . . . . . . . . . . . . . 14 (𝑗 ∈ ℕ0 → ((𝑁𝑆) ∈ ℕ → (𝑗 < (𝑁𝑆) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))))
43423imp 1105 . . . . . . . . . . . . 13 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4426, 43sylbi 218 . . . . . . . . . . . 12 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4544impcom 408 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) < 𝑁)
46 elfzo0 13068 . . . . . . . . . . 11 ((𝑗 + 𝑆) ∈ (0..^𝑁) ↔ ((𝑗 + 𝑆) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑗 + 𝑆) < 𝑁))
4719, 25, 45, 46syl3anbrc 1337 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0..^𝑁))
4847adantr 481 . . . . . . . . 9 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑗 + 𝑆) ∈ (0..^𝑁))
49 fveq2 6667 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
5049adantl 482 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
51 fvoveq1 7171 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃‘(𝑖 + 1)) = (𝑃‘((𝑗 + 𝑆) + 1)))
52 simpr 485 . . . . . . . . . . . . 13 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
5352fveq2d 6671 . . . . . . . . . . . 12 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑃‘((𝑗 + 𝑆) + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
5451, 53sylan9eqr 2883 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝑃‘(𝑖 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
5550, 54eqeq12d 2842 . . . . . . . . . 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))))
56 2fveq3 6672 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
5749sneqd 4576 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → {(𝑃𝑖)} = {(𝑃‘(𝑗 + 𝑆))})
5856, 57eqeq12d 2842 . . . . . . . . . . 11 (𝑖 = (𝑗 + 𝑆) → ((𝐼‘(𝐹𝑖)) = {(𝑃𝑖)} ↔ (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}))
5958adantl 482 . . . . . . . . . 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))}))
6050, 54preq12d 4676 . . . . . . . . . . 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))})
6156adantl 482 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
6260, 61sseq12d 4004 . . . . . . . . . 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)))))
6355, 59, 62ifpbi123d 1071 . . . . . . . . 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))))))
6448, 63rspcdv 3619 . . . . . . . 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))))))
6510, 64mpdan 683 . . . . . . 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))))))
662, 65sylan 580 . . . . . 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))))))
6766ex 413 . . . . 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)))))))
681, 67mpid 44 . . . 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))))))
6968imp 407 . . 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)))))
70 elfzofz 13043 . . . . 5 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ (0...(𝑁𝑆)))
71 crctcshwlkn0lem.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
722, 71crctcshwlkn0lem2 27503 . . . . 5 ((𝜑𝑗 ∈ (0...(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
7370, 72sylan2 592 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
74 fzofzp1 13124 . . . . 5 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 1) ∈ (0...(𝑁𝑆)))
752, 71crctcshwlkn0lem2 27503 . . . . 5 ((𝜑 ∧ (𝑗 + 1) ∈ (0...(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
7674, 75sylan2 592 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
77 crctcshwlkn0lem.h . . . . . . 7 𝐻 = (𝐹 cyclShift 𝑆)
7877fveq1i 6668 . . . . . 6 (𝐻𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗)
79 crctcshwlkn0lem.f . . . . . . . . 9 (𝜑𝐹 ∈ Word 𝐴)
8079adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝐹 ∈ Word 𝐴)
812, 6syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ ℤ)
8281adantr 481 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℤ)
83 nnz 11993 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
8483adantl 482 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
85 nnz 11993 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
8685adantr 481 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ ℤ)
8784, 86zsubcld 12081 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
8813nn0ge0d 11947 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 0 ≤ 𝑆)
8988adantr 481 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑆)
90 subge02 11145 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9129, 28, 90syl2anr 596 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9289, 91mpbid 233 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ≤ 𝑁)
9387, 84, 923jca 1122 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
94933adant3 1126 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
9511, 94sylbi 218 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
96 eluz2 12238 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘(𝑁𝑆)) ↔ ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
9795, 96sylibr 235 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘(𝑁𝑆)))
98 fzoss2 13055 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘(𝑁𝑆)) → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
992, 97, 983syl 18 . . . . . . . . . 10 (𝜑 → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
10099sselda 3971 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^𝑁))
101 crctcshwlkn0lem.n . . . . . . . . . 10 𝑁 = (♯‘𝐹)
102101oveq2i 7159 . . . . . . . . 9 (0..^𝑁) = (0..^(♯‘𝐹))
103100, 102syl6eleq 2928 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^(♯‘𝐹)))
104 cshwidxmod 14155 . . . . . . . 8 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
10580, 82, 103, 104syl3anc 1365 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
106101eqcomi 2835 . . . . . . . . . 10 (♯‘𝐹) = 𝑁
107106oveq2i 7159 . . . . . . . . 9 ((𝑗 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod 𝑁)
10817imp 407 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
109 nnm1nn0 11927 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1101093ad2ant2 1128 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 − 1) ∈ ℕ0)
111110adantr 481 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑁 − 1) ∈ ℕ0)
11227, 31anim12i 612 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
113112, 34sylibr 235 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
114113, 37syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 < (𝑁𝑆) ↔ (𝑗 + 𝑆) < 𝑁))
115133ad2ant1 1127 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℕ0)
116115, 14sylan2 592 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℕ0)
117116nn0zd 12074 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℤ)
118833ad2ant2 1128 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℤ)
119118adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → 𝑁 ∈ ℤ)
120 zltlem1 12024 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
121117, 119, 120syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
122121biimpd 230 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
123114, 122sylbid 241 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
124123impancom 452 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
1251243adant2 1125 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
12626, 125sylbi 218 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
127126impcom 408 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ≤ (𝑁 − 1))
128108, 111, 1273jca 1122 . . . . . . . . . . . . 13 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
12911, 128sylanb 581 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
130 elfz2nn0 12988 . . . . . . . . . . . 12 ((𝑗 + 𝑆) ∈ (0...(𝑁 − 1)) ↔ ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
131129, 130sylibr 235 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0...(𝑁 − 1)))
132 zaddcl 12011 . . . . . . . . . . . . 13 ((𝑗 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑗 + 𝑆) ∈ ℤ)
1333, 6, 132syl2anr 596 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℤ)
134 zmodid2 13257 . . . . . . . . . . . 12 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
135133, 25, 134syl2anc 584 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
136131, 135mpbird 258 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
1372, 136sylan 580 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
138107, 137syl5eq 2873 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod (♯‘𝐹)) = (𝑗 + 𝑆))
139138fveq2d 6671 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) = (𝐹‘(𝑗 + 𝑆)))
140105, 139eqtrd 2861 . . . . . 6 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘(𝑗 + 𝑆)))
14178, 140syl5eq 2873 . . . . 5 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐻𝑗) = (𝐹‘(𝑗 + 𝑆)))
142141fveq2d 6671 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
143 simp1 1130 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
144 simp2 1131 . . . . . 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)))
145143, 144eqeq12d 2842 . . . . 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))))
146 simp3 1132 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
147143sneqd 4576 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → {(𝑄𝑗)} = {(𝑃‘(𝑗 + 𝑆))})
148146, 147eqeq12d 2842 . . . . 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))}))
149143, 144preq12d 4676 . . . . . 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))})
150149, 146sseq12d 4004 . . . . 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)))))
151145, 148, 150ifpbi123d 1071 . . . 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))))))
15273, 76, 142, 151syl3anc 1365 . . 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))))))
15369, 152mpbird 258 . 2 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
154153ralrimiva 3187 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 207  wa 396  if-wif 1056  w3a 1081   = wceq 1530  wcel 2107  wral 3143  wss 3940  ifcif 4470  {csn 4564  {cpr 4566   class class class wbr 5063  cmpt 5143  cfv 6352  (class class class)co 7148  cc 10524  cr 10525  0cc0 10526  1c1 10527   + caddc 10529   < clt 10664  cle 10665  cmin 10859  cn 11627  0cn0 11886  cz 11970  cuz 12232  ...cfz 12882  ..^cfzo 13023   mod cmo 13227  chash 13680  Word cword 13851   cyclShift ccsh 14140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-pre-sup 10604
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-nel 3129  df-ral 3148  df-rex 3149  df-reu 3150  df-rmo 3151  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-int 4875  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-oadd 8097  df-er 8279  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-sup 8895  df-inf 8896  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-n0 11887  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-fl 13152  df-mod 13228  df-hash 13681  df-word 13852  df-concat 13913  df-substr 13993  df-pfx 14023  df-csh 14141
This theorem is referenced by:  crctcshwlkn0lem7  27508
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