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Theorem cshf1 13602
 Description: Cyclically shifting a word which contains a symbol at most once results in a word which contains a symbol at most once. (Contributed by AV, 14-Mar-2021.)
Assertion
Ref Expression
cshf1 ((𝐹:(0..^(#‘𝐹))–1-1𝐴𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(#‘𝐹))–1-1𝐴)

Proof of Theorem cshf1
Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1f 6139 . . . . 5 (𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹:(0..^(#‘𝐹))⟶𝐴)
2 iswrdi 13341 . . . . 5 (𝐹:(0..^(#‘𝐹))⟶𝐴𝐹 ∈ Word 𝐴)
31, 2syl 17 . . . 4 (𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴)
4 cshwf 13592 . . . . . . . . 9 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
543adant1 1099 . . . . . . . 8 ((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
65adantr 480 . . . . . . 7 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
7 feq1 6064 . . . . . . . 8 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴))
87adantl 481 . . . . . . 7 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴))
96, 8mpbird 247 . . . . . 6 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(#‘𝐹))⟶𝐴)
10 dff13 6552 . . . . . . . 8 (𝐹:(0..^(#‘𝐹))–1-1𝐴 ↔ (𝐹:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
11 fveq1 6228 . . . . . . . . . . . . . . . . . 18 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
12113ad2ant1 1102 . . . . . . . . . . . . . . . . 17 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐺𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
1312adantr 480 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
14 cshwidxmod 13595 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
15143expia 1286 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹)))))
16153adant1 1099 . . . . . . . . . . . . . . . . . . 19 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹)))))
1716com12 32 . . . . . . . . . . . . . . . . . 18 (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹)))))
1817adantr 480 . . . . . . . . . . . . . . . . 17 ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹)))))
1918impcom 445 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
2013, 19eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
21 fveq1 6228 . . . . . . . . . . . . . . . . . 18 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
22213ad2ant1 1102 . . . . . . . . . . . . . . . . 17 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐺𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
2322adantr 480 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
24 cshwidxmod 13595 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
25243expia 1286 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
26253adant1 1099 . . . . . . . . . . . . . . . . . 18 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
2726adantld 482 . . . . . . . . . . . . . . . . 17 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
2827imp 444 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
2923, 28eqtrd 2685 . . . . . . . . . . . . . . 15 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
3020, 29eqeq12d 2666 . . . . . . . . . . . . . 14 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝐺𝑖) = (𝐺𝑗) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
3130adantlr 751 . . . . . . . . . . . . 13 ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝐺𝑖) = (𝐺𝑗) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
32 elfzo0 12548 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^(#‘𝐹)) ↔ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑖 < (#‘𝐹)))
33 nn0z 11438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑖 ∈ ℕ0𝑖 ∈ ℤ)
3433adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → 𝑖 ∈ ℤ)
3534adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑖 ∈ ℤ)
36 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
3735, 36zaddcld 11524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (𝑖 + 𝑆) ∈ ℤ)
38 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → (#‘𝐹) ∈ ℕ)
3938adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (#‘𝐹) ∈ ℕ)
4037, 39jca 553 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
4140ex 449 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑆 ∈ ℤ → ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
42413ad2ant3 1104 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4342com12 32 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
44433adant3 1101 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑖 < (#‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4532, 44sylbi 207 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4645adantr 480 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4746impcom 445 . . . . . . . . . . . . . . . . . 18 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
48 zmodfzo 12733 . . . . . . . . . . . . . . . . . 18 (((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) mod (#‘𝐹)) ∈ (0..^(#‘𝐹)))
4947, 48syl 17 . . . . . . . . . . . . . . . . 17 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑖 + 𝑆) mod (#‘𝐹)) ∈ (0..^(#‘𝐹)))
50 elfzo0 12548 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0..^(#‘𝐹)) ↔ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑗 < (#‘𝐹)))
51 nn0z 11438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ℕ0𝑗 ∈ ℤ)
5251adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → 𝑗 ∈ ℤ)
5352adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑗 ∈ ℤ)
54 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
5553, 54zaddcld 11524 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (𝑗 + 𝑆) ∈ ℤ)
56 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → (#‘𝐹) ∈ ℕ)
5756adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (#‘𝐹) ∈ ℕ)
5855, 57jca 553 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
5958expcom 450 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
60593adant3 1101 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑗 < (#‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6150, 60sylbi 207 . . . . . . . . . . . . . . . . . . . . . 22 (𝑗 ∈ (0..^(#‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6261com12 32 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℤ → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
63623ad2ant3 1104 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6463adantld 482 . . . . . . . . . . . . . . . . . . 19 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6564imp 444 . . . . . . . . . . . . . . . . . 18 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
66 zmodfzo 12733 . . . . . . . . . . . . . . . . . 18 (((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ) → ((𝑗 + 𝑆) mod (#‘𝐹)) ∈ (0..^(#‘𝐹)))
6765, 66syl 17 . . . . . . . . . . . . . . . . 17 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑗 + 𝑆) mod (#‘𝐹)) ∈ (0..^(#‘𝐹)))
68 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (𝐹𝑥) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
6968eqeq1d 2653 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦)))
70 eqeq1 2655 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (𝑥 = 𝑦 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦))
7169, 70imbi12d 333 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦) → ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦)))
72 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (𝐹𝑦) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
7372eqeq2d 2661 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
74 eqeq2 2662 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
7573, 74imbi12d 333 . . . . . . . . . . . . . . . . . 18 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦) → ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦) ↔ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
7671, 75rspc2v 3353 . . . . . . . . . . . . . . . . 17 ((((𝑖 + 𝑆) mod (#‘𝐹)) ∈ (0..^(#‘𝐹)) ∧ ((𝑗 + 𝑆) mod (#‘𝐹)) ∈ (0..^(#‘𝐹))) → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
7749, 67, 76syl2anc 694 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
78 simpr 476 . . . . . . . . . . . . . . . . . 18 ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) ∧ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
79 addmodlteq 12785 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
80793expa 1284 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
8180ancoms 468 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
8281bicomd 213 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
8382ex 449 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℤ → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
84833ad2ant3 1104 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
8584imp 444 . . . . . . . . . . . . . . . . . . 19 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
8685adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) ∧ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
8778, 86sylibrd 249 . . . . . . . . . . . . . . . . 17 ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) ∧ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → 𝑖 = 𝑗))
8887ex 449 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → 𝑖 = 𝑗)))
8977, 88syld 47 . . . . . . . . . . . . . . 15 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → 𝑖 = 𝑗)))
9089impancom 455 . . . . . . . . . . . . . 14 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → 𝑖 = 𝑗)))
9190imp 444 . . . . . . . . . . . . 13 ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))) → 𝑖 = 𝑗))
9231, 91sylbid 230 . . . . . . . . . . . 12 ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗))
9392ralrimivva 3000 . . . . . . . . . . 11 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) → ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗))
94933exp1 1305 . . . . . . . . . 10 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))))
9594com14 96 . . . . . . . . 9 (∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))))
9695adantl 481 . . . . . . . 8 ((𝐹:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)) → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))))
9710, 96sylbi 207 . . . . . . 7 (𝐹:(0..^(#‘𝐹))–1-1𝐴 → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))))
98973imp1 1302 . . . . . 6 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗))
999, 98jca 553 . . . . 5 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))
100993exp1 1305 . . . 4 (𝐹:(0..^(#‘𝐹))–1-1𝐴 → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗))))))
1013, 100mpd 15 . . 3 (𝐹:(0..^(#‘𝐹))–1-1𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))))
1021013imp 1275 . 2 ((𝐹:(0..^(#‘𝐹))–1-1𝐴𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))
103 dff13 6552 . 2 (𝐺:(0..^(#‘𝐹))–1-1𝐴 ↔ (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))
104102, 103sylibr 224 1 ((𝐹:(0..^(#‘𝐹))–1-1𝐴𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(#‘𝐹))–1-1𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941   class class class wbr 4685  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  (class class class)co 6690  0cc0 9974   + caddc 9977   < clt 10112  ℕcn 11058  ℕ0cn0 11330  ℤcz 11415  ..^cfzo 12504   mod cmo 12708  #chash 13157  Word cword 13323   cyclShift ccsh 13580 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-rp 11871  df-fz 12365  df-fzo 12505  df-fl 12633  df-mod 12709  df-hash 13158  df-word 13331  df-concat 13333  df-substr 13335  df-csh 13581 This theorem is referenced by:  cshinj  13603
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