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Theorem crctcshwlkn0lem4 29332
Description: Lemma for crctcshwlkn0 29340. (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 13638 . . . . . . . . . . 11 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℤ)
43zcnd 12673 . . . . . . . . . 10 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℂ)
54adantl 480 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ ℂ)
6 elfzoelz 13638 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℤ)
76zcnd 12673 . . . . . . . . . 10 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ ℂ)
87adantr 479 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℂ)
9 1cnd 11215 . . . . . . . . 9 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 1 ∈ ℂ)
105, 8, 9add32d 11447 . . . . . . . 8 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
11 elfzo1 13688 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
12 elfzonn0 13683 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → 𝑗 ∈ ℕ0)
13 nnnn0 12485 . . . . . . . . . . . . . . 15 (𝑆 ∈ ℕ → 𝑆 ∈ ℕ0)
14 nn0addcl 12513 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ0𝑆 ∈ ℕ0) → (𝑗 + 𝑆) ∈ ℕ0)
1514ex 411 . . . . . . . . . . . . . . 15 (𝑗 ∈ ℕ0 → (𝑆 ∈ ℕ0 → (𝑗 + 𝑆) ∈ ℕ0))
1612, 13, 15syl2imc 41 . . . . . . . . . . . . . 14 (𝑆 ∈ ℕ → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
17163ad2ant1 1131 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
1811, 17sylbi 216 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 𝑆) ∈ ℕ0))
1918imp 405 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
20 fzo0ss1 13668 . . . . . . . . . . . . . 14 (1..^𝑁) ⊆ (0..^𝑁)
2120sseli 3979 . . . . . . . . . . . . 13 (𝑆 ∈ (1..^𝑁) → 𝑆 ∈ (0..^𝑁))
22 elfzo0 13679 . . . . . . . . . . . . . 14 (𝑆 ∈ (0..^𝑁) ↔ (𝑆 ∈ ℕ0𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
2322simp2bi 1144 . . . . . . . . . . . . 13 (𝑆 ∈ (0..^𝑁) → 𝑁 ∈ ℕ)
2421, 23syl 17 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ ℕ)
2524adantr 479 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → 𝑁 ∈ ℕ)
26 elfzo0 13679 . . . . . . . . . . . . 13 (𝑗 ∈ (0..^(𝑁𝑆)) ↔ (𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)))
27 nn0re 12487 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℕ0𝑗 ∈ ℝ)
28 nnre 12225 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
29 nnre 12225 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
3028, 29anim12i 611 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
31303adant3 1130 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3211, 31sylbi 216 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ (1..^𝑁) → (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
3327, 32anim12i 611 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
34 3anass 1093 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) ↔ (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
3533, 34sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ))
36 ltaddsub 11694 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℝ ∧ 𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → ((𝑗 + 𝑆) < 𝑁𝑗 < (𝑁𝑆)))
3736bicomd 222 . . . . . . . . . . . . . . . . . . 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 228 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑆 ∈ (1..^𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) < 𝑁))
4039ex 411 . . . . . . . . . . . . . . . 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 1109 . . . . . . . . . . . . 13 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4426, 43sylbi 216 . . . . . . . . . . . 12 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑆 ∈ (1..^𝑁) → (𝑗 + 𝑆) < 𝑁))
4544impcom 406 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) < 𝑁)
46 elfzo0 13679 . . . . . . . . . . 11 ((𝑗 + 𝑆) ∈ (0..^𝑁) ↔ ((𝑗 + 𝑆) ∈ ℕ0𝑁 ∈ ℕ ∧ (𝑗 + 𝑆) < 𝑁))
4719, 25, 45, 46syl3anbrc 1341 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0..^𝑁))
4847adantr 479 . . . . . . . . 9 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑗 + 𝑆) ∈ (0..^𝑁))
49 fveq2 6892 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
5049adantl 480 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝑃𝑖) = (𝑃‘(𝑗 + 𝑆)))
51 fvoveq1 7436 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝑃‘(𝑖 + 1)) = (𝑃‘((𝑗 + 𝑆) + 1)))
52 simpr 483 . . . . . . . . . . . . 13 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆))
5352fveq2d 6896 . . . . . . . . . . . 12 (((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) → (𝑃‘((𝑗 + 𝑆) + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
5451, 53sylan9eqr 2792 . . . . . . . . . . 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 2746 . . . . . . . . . 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 6897 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
5749sneqd 4641 . . . . . . . . . . . 12 (𝑖 = (𝑗 + 𝑆) → {(𝑃𝑖)} = {(𝑃‘(𝑗 + 𝑆))})
5856, 57eqeq12d 2746 . . . . . . . . . . 11 (𝑖 = (𝑗 + 𝑆) → ((𝐼‘(𝐹𝑖)) = {(𝑃𝑖)} ↔ (𝐼‘(𝐹‘(𝑗 + 𝑆))) = {(𝑃‘(𝑗 + 𝑆))}))
5958adantl 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))}))
6050, 54preq12d 4746 . . . . . . . . . . 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 480 . . . . . . . . . . 11 ((((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) ∧ ((𝑗 + 𝑆) + 1) = ((𝑗 + 1) + 𝑆)) ∧ 𝑖 = (𝑗 + 𝑆)) → (𝐼‘(𝐹𝑖)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
6260, 61sseq12d 4016 . . . . . . . . . 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 1076 . . . . . . . . 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 3605 . . . . . . . 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 578 . . . . . 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 411 . . . . 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 405 . . 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 13654 . . . . 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 29330 . . . . 5 ((𝜑𝑗 ∈ (0...(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
7370, 72sylan2 591 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
74 fzofzp1 13735 . . . . 5 (𝑗 ∈ (0..^(𝑁𝑆)) → (𝑗 + 1) ∈ (0...(𝑁𝑆)))
752, 71crctcshwlkn0lem2 29330 . . . . 5 ((𝜑 ∧ (𝑗 + 1) ∈ (0...(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
7674, 75sylan2 591 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)))
77 crctcshwlkn0lem.h . . . . . . 7 𝐻 = (𝐹 cyclShift 𝑆)
7877fveq1i 6893 . . . . . 6 (𝐻𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗)
79 crctcshwlkn0lem.f . . . . . . . . 9 (𝜑𝐹 ∈ Word 𝐴)
8079adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝐹 ∈ Word 𝐴)
812, 6syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ ℤ)
8281adantr 479 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑆 ∈ ℤ)
83 nnz 12585 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
8483adantl 480 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℤ)
85 nnz 12585 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
8685adantr 479 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑆 ∈ ℤ)
8784, 86zsubcld 12677 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
8813nn0ge0d 12541 . . . . . . . . . . . . . . . . 17 (𝑆 ∈ ℕ → 0 ≤ 𝑆)
8988adantr 479 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 0 ≤ 𝑆)
90 subge02 11736 . . . . . . . . . . . . . . . . 17 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9129, 28, 90syl2anr 595 . . . . . . . . . . . . . . . 16 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (0 ≤ 𝑆 ↔ (𝑁𝑆) ≤ 𝑁))
9289, 91mpbid 231 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ≤ 𝑁)
9387, 84, 923jca 1126 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
94933adant3 1130 . . . . . . . . . . . . 13 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
9511, 94sylbi 216 . . . . . . . . . . . 12 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
96 eluz2 12834 . . . . . . . . . . . 12 (𝑁 ∈ (ℤ‘(𝑁𝑆)) ↔ ((𝑁𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ (𝑁𝑆) ≤ 𝑁))
9795, 96sylibr 233 . . . . . . . . . . 11 (𝑆 ∈ (1..^𝑁) → 𝑁 ∈ (ℤ‘(𝑁𝑆)))
98 fzoss2 13666 . . . . . . . . . . 11 (𝑁 ∈ (ℤ‘(𝑁𝑆)) → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
992, 97, 983syl 18 . . . . . . . . . 10 (𝜑 → (0..^(𝑁𝑆)) ⊆ (0..^𝑁))
10099sselda 3983 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^𝑁))
101 crctcshwlkn0lem.n . . . . . . . . . 10 𝑁 = (♯‘𝐹)
102101oveq2i 7424 . . . . . . . . 9 (0..^𝑁) = (0..^(♯‘𝐹))
103100, 102eleqtrdi 2841 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → 𝑗 ∈ (0..^(♯‘𝐹)))
104 cshwidxmod 14759 . . . . . . . 8 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
10580, 82, 103, 104syl3anc 1369 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
106101eqcomi 2739 . . . . . . . . . 10 (♯‘𝐹) = 𝑁
107106oveq2i 7424 . . . . . . . . 9 ((𝑗 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod 𝑁)
10817imp 405 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℕ0)
109 nnm1nn0 12519 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ0)
1101093ad2ant2 1132 . . . . . . . . . . . . . . 15 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁 − 1) ∈ ℕ0)
111110adantr 479 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑁 − 1) ∈ ℕ0)
11227, 31anim12i 611 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 ∈ ℝ ∧ (𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ)))
113112, 34sylibr 233 . . . . . . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑆 ∈ ℕ0)
116115, 14sylan2 591 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℕ0)
117116nn0zd 12590 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 + 𝑆) ∈ ℤ)
118833ad2ant2 1132 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℤ)
119118adantl 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → 𝑁 ∈ ℤ)
120 zltlem1 12621 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
121117, 119, 120syl2anc 582 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 ↔ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
122121biimpd 228 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → ((𝑗 + 𝑆) < 𝑁 → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
123114, 122sylbid 239 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ ℕ0 ∧ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁)) → (𝑗 < (𝑁𝑆) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
124123impancom 450 . . . . . . . . . . . . . . . . 17 ((𝑗 ∈ ℕ0𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
1251243adant2 1129 . . . . . . . . . . . . . . . 16 ((𝑗 ∈ ℕ0 ∧ (𝑁𝑆) ∈ ℕ ∧ 𝑗 < (𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
12626, 125sylbi 216 . . . . . . . . . . . . . . 15 (𝑗 ∈ (0..^(𝑁𝑆)) → ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑗 + 𝑆) ≤ (𝑁 − 1)))
127126impcom 406 . . . . . . . . . . . . . 14 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ≤ (𝑁 − 1))
128108, 111, 1273jca 1126 . . . . . . . . . . . . 13 (((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
12911, 128sylanb 579 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
130 elfz2nn0 13598 . . . . . . . . . . . 12 ((𝑗 + 𝑆) ∈ (0...(𝑁 − 1)) ↔ ((𝑗 + 𝑆) ∈ ℕ0 ∧ (𝑁 − 1) ∈ ℕ0 ∧ (𝑗 + 𝑆) ≤ (𝑁 − 1)))
131129, 130sylibr 233 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ (0...(𝑁 − 1)))
132 zaddcl 12608 . . . . . . . . . . . . 13 ((𝑗 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑗 + 𝑆) ∈ ℤ)
1333, 6, 132syl2anr 595 . . . . . . . . . . . 12 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (𝑗 + 𝑆) ∈ ℤ)
134 zmodid2 13870 . . . . . . . . . . . 12 (((𝑗 + 𝑆) ∈ ℤ ∧ 𝑁 ∈ ℕ) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
135133, 25, 134syl2anc 582 . . . . . . . . . . 11 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → (((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆) ↔ (𝑗 + 𝑆) ∈ (0...(𝑁 − 1))))
136131, 135mpbird 256 . . . . . . . . . 10 ((𝑆 ∈ (1..^𝑁) ∧ 𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
1372, 136sylan 578 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod 𝑁) = (𝑗 + 𝑆))
138107, 137eqtrid 2782 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝑗 + 𝑆) mod (♯‘𝐹)) = (𝑗 + 𝑆))
139138fveq2d 6896 . . . . . . 7 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) = (𝐹‘(𝑗 + 𝑆)))
140105, 139eqtrd 2770 . . . . . 6 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘(𝑗 + 𝑆)))
14178, 140eqtrid 2782 . . . . 5 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐻𝑗) = (𝐹‘(𝑗 + 𝑆)))
142141fveq2d 6896 . . . 4 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆))))
143 simp1 1134 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → (𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)))
144 simp2 1135 . . . . . 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 2746 . . . . 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 1136 . . . . . 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 4641 . . . . . 6 (((𝑄𝑗) = (𝑃‘(𝑗 + 𝑆)) ∧ (𝑄‘(𝑗 + 1)) = (𝑃‘((𝑗 + 1) + 𝑆)) ∧ (𝐼‘(𝐻𝑗)) = (𝐼‘(𝐹‘(𝑗 + 𝑆)))) → {(𝑄𝑗)} = {(𝑃‘(𝑗 + 𝑆))})
148146, 147eqeq12d 2746 . . . . 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 4746 . . . . . 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 4016 . . . . 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 1076 . . . 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 1369 . . 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 256 . 2 ((𝜑𝑗 ∈ (0..^(𝑁𝑆))) → if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
154153ralrimiva 3144 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 205  wa 394  if-wif 1059  w3a 1085   = wceq 1539  wcel 2104  wral 3059  wss 3949  ifcif 4529  {csn 4629  {cpr 4631   class class class wbr 5149  cmpt 5232  cfv 6544  (class class class)co 7413  cc 11112  cr 11113  0cc0 11114  1c1 11115   + caddc 11117   < clt 11254  cle 11255  cmin 11450  cn 12218  0cn0 12478  cz 12564  cuz 12828  ...cfz 13490  ..^cfzo 13633   mod cmo 13840  chash 14296  Word cword 14470   cyclShift ccsh 14744
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-sup 9441  df-inf 9442  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-div 11878  df-nn 12219  df-2 12281  df-n0 12479  df-z 12565  df-uz 12829  df-rp 12981  df-fz 13491  df-fzo 13634  df-fl 13763  df-mod 13841  df-hash 14297  df-word 14471  df-concat 14527  df-substr 14597  df-pfx 14627  df-csh 14745
This theorem is referenced by:  crctcshwlkn0lem7  29335
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