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Theorem cshf1 13349
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 5995 . . . . 5 (𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹:(0..^(#‘𝐹))⟶𝐴)
2 iswrdi 13106 . . . . 5 (𝐹:(0..^(#‘𝐹))⟶𝐴𝐹 ∈ Word 𝐴)
31, 2syl 17 . . . 4 (𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴)
4 cshwf 13339 . . . . . . . . 9 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
543adant1 1071 . . . . . . . 8 ((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
65adantr 479 . . . . . . 7 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
7 feq1 5921 . . . . . . . 8 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(#‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴))
87adantl 480 . . . . . . 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 245 . . . . . 6 (((𝐹:(0..^(#‘𝐹))–1-1𝐴𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(#‘𝐹))⟶𝐴)
10 dff13 6390 . . . . . . . 8 (𝐹:(0..^(#‘𝐹))–1-1𝐴 ↔ (𝐹:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑥 ∈ (0..^(#‘𝐹))∀𝑦 ∈ (0..^(#‘𝐹))((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦)))
11 fveq1 6083 . . . . . . . . . . . . . . . . . 18 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
12113ad2ant1 1074 . . . . . . . . . . . . . . . . 17 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐺𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
1312adantr 479 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
14 cshwidxmod 13342 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
15143expia 1258 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹)))))
16153adant1 1071 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . 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 2639 . . . . . . . . . . . . . . 15 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
21 fveq1 6083 . . . . . . . . . . . . . . . . . 18 (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
22213ad2ant1 1074 . . . . . . . . . . . . . . . . 17 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝐺𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
2322adantr 479 . . . . . . . . . . . . . . . 16 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
24 cshwidxmod 13342 . . . . . . . . . . . . . . . . . . . 20 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
25243expia 1258 . . . . . . . . . . . . . . . . . . 19 ((𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
26253adant1 1071 . . . . . . . . . . . . . . . . . 18 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
2726adantld 481 . . . . . . . . . . . . . . . . 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 443 . . . . . . . . . . . . . . . 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 2639 . . . . . . . . . . . . . . 15 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝐺𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
3020, 29eqeq12d 2620 . . . . . . . . . . . . . 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 746 . . . . . . . . . . . . 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 12327 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (0..^(#‘𝐹)) ↔ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑖 < (#‘𝐹)))
33 nn0z 11229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑖 ∈ ℕ0𝑖 ∈ ℤ)
3433adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → 𝑖 ∈ ℤ)
3534adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑖 ∈ ℤ)
36 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
3735, 36zaddcld 11314 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (𝑖 + 𝑆) ∈ ℤ)
38 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → (#‘𝐹) ∈ ℕ)
3938adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (#‘𝐹) ∈ ℕ)
4037, 39jca 552 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
4140ex 448 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑆 ∈ ℤ → ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
42413ad2ant3 1076 . . . . . . . . . . . . . . . . . . . . . . 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 1073 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑖 < (#‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4532, 44sylbi 205 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4645adantr 479 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
4746impcom 444 . . . . . . . . . . . . . . . . . 18 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
48 zmodfzo 12506 . . . . . . . . . . . . . . . . . 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 12327 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗 ∈ (0..^(#‘𝐹)) ↔ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑗 < (#‘𝐹)))
51 nn0z 11229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∈ ℕ0𝑗 ∈ ℤ)
5251adantr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → 𝑗 ∈ ℤ)
5352adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑗 ∈ ℤ)
54 simpl 471 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
5553, 54zaddcld 11314 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (𝑗 + 𝑆) ∈ ℤ)
56 simpr 475 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → (#‘𝐹) ∈ ℕ)
5756adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → (#‘𝐹) ∈ ℕ)
5855, 57jca 552 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
5958expcom 449 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
60593adant3 1073 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑗 ∈ ℕ0 ∧ (#‘𝐹) ∈ ℕ ∧ 𝑗 < (#‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6150, 60sylbi 205 . . . . . . . . . . . . . . . . . . . . . 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 1076 . . . . . . . . . . . . . . . . . . . 20 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6463adantld 481 . . . . . . . . . . . . . . . . . . 19 ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
6564imp 443 . . . . . . . . . . . . . . . . . 18 (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
66 zmodfzo 12506 . . . . . . . . . . . . . . . . . 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 6084 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (𝐹𝑥) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
6968eqeq1d 2607 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → ((𝐹𝑥) = (𝐹𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦)))
70 eqeq1 2609 . . . . . . . . . . . . . . . . . . 19 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (𝑥 = 𝑦 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦))
7169, 70imbi12d 332 . . . . . . . . . . . . . . . . . 18 (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (((𝐹𝑥) = (𝐹𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦) → ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦)))
72 fveq2 6084 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (𝐹𝑦) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
7372eqeq2d 2615 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
74 eqeq2 2616 . . . . . . . . . . . . . . . . . . 19 (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
7573, 74imbi12d 332 . . . . . . . . . . . . . . . . . 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 3288 . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . 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 475 . . . . . . . . . . . . . . . . . 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 12558 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
80793expa 1256 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
8180ancoms 467 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
8281bicomd 211 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
8382ex 448 . . . . . . . . . . . . . . . . . . . . 21 (𝑆 ∈ ℤ → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
84833ad2ant3 1076 . . . . . . . . . . . . . . . . . . . 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 443 . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . . . 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 448 . . . . . . . . . . . . . . . 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 45 . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . 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 443 . . . . . . . . . . . . 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 228 . . . . . . . . . . . 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 2949 . . . . . . . . . . 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 1274 . . . . . . . . . 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 93 . . . . . . . . 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 480 . . . . . . . 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 205 . . . . . . 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 1271 . . . . . 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 552 . . . . 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 1274 . . . 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 1248 . 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 6390 . 2 (𝐺:(0..^(#‘𝐹))–1-1𝐴 ↔ (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺𝑖) = (𝐺𝑗) → 𝑖 = 𝑗)))
104102, 103sylibr 222 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  wral 2891   class class class wbr 4573  wf 5782  1-1wf1 5783  cfv 5786  (class class class)co 6523  0cc0 9788   + caddc 9791   < clt 9926  cn 10863  0cn0 11135  cz 11206  ..^cfzo 12285   mod cmo 12481  #chash 12930  Word cword 13088   cyclShift ccsh 13327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820  ax-cnex 9844  ax-resscn 9845  ax-1cn 9846  ax-icn 9847  ax-addcl 9848  ax-addrcl 9849  ax-mulcl 9850  ax-mulrcl 9851  ax-mulcom 9852  ax-addass 9853  ax-mulass 9854  ax-distr 9855  ax-i2m1 9856  ax-1ne0 9857  ax-1rid 9858  ax-rnegex 9859  ax-rrecex 9860  ax-cnre 9861  ax-pre-lttri 9862  ax-pre-lttrn 9863  ax-pre-ltadd 9864  ax-pre-mulgt0 9865  ax-pre-sup 9866
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-nel 2778  df-ral 2896  df-rex 2897  df-reu 2898  df-rmo 2899  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-int 4401  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-pred 5579  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-riota 6485  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-om 6931  df-1st 7032  df-2nd 7033  df-wrecs 7267  df-recs 7328  df-rdg 7366  df-1o 7420  df-oadd 7424  df-er 7602  df-en 7815  df-dom 7816  df-sdom 7817  df-fin 7818  df-sup 8204  df-inf 8205  df-card 8621  df-pnf 9928  df-mnf 9929  df-xr 9930  df-ltxr 9931  df-le 9932  df-sub 10115  df-neg 10116  df-div 10530  df-nn 10864  df-2 10922  df-n0 11136  df-z 11207  df-uz 11516  df-rp 11661  df-fz 12149  df-fzo 12286  df-fl 12406  df-mod 12482  df-hash 12931  df-word 13096  df-concat 13098  df-substr 13100  df-csh 13328
This theorem is referenced by:  cshinj  13350
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