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Theorem cshimadifsn0 13370
Description: The image of a cyclically shifted word under its domain without its upper bound is the image of a cyclically shifted word under its domain without the number of shifted symbols. (Contributed by AV, 19-Mar-2021.)
Assertion
Ref Expression
cshimadifsn0 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))

Proof of Theorem cshimadifsn0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cshimadifsn 13369 . 2 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
2 elfzoel2 12290 . . . . . . . 8 (𝐽 ∈ (0..^𝑁) → 𝑁 ∈ ℤ)
3 elfzom1elp1fzo1 12386 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (1..^𝑁))
43ex 448 . . . . . . . 8 (𝑁 ∈ ℤ → (𝑦 ∈ (0..^(𝑁 − 1)) → (𝑦 + 1) ∈ (1..^𝑁)))
52, 4syl 17 . . . . . . 7 (𝐽 ∈ (0..^𝑁) → (𝑦 ∈ (0..^(𝑁 − 1)) → (𝑦 + 1) ∈ (1..^𝑁)))
653ad2ant3 1076 . . . . . 6 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (0..^(𝑁 − 1)) → (𝑦 + 1) ∈ (1..^𝑁)))
76imp 443 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (1..^𝑁))
8 elfzo1elm1fzo0 12387 . . . . . . 7 (𝑥 ∈ (1..^𝑁) → (𝑥 − 1) ∈ (0..^(𝑁 − 1)))
98adantl 480 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) → (𝑥 − 1) ∈ (0..^(𝑁 − 1)))
10 oveq1 6531 . . . . . . . 8 (𝑦 = (𝑥 − 1) → (𝑦 + 1) = ((𝑥 − 1) + 1))
1110eqeq2d 2616 . . . . . . 7 (𝑦 = (𝑥 − 1) → (𝑥 = (𝑦 + 1) ↔ 𝑥 = ((𝑥 − 1) + 1)))
1211adantl 480 . . . . . 6 ((((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) ∧ 𝑦 = (𝑥 − 1)) → (𝑥 = (𝑦 + 1) ↔ 𝑥 = ((𝑥 − 1) + 1)))
13 elfzoelz 12291 . . . . . . . . . 10 (𝑥 ∈ (1..^𝑁) → 𝑥 ∈ ℤ)
1413zcnd 11312 . . . . . . . . 9 (𝑥 ∈ (1..^𝑁) → 𝑥 ∈ ℂ)
15 npcan1 10303 . . . . . . . . 9 (𝑥 ∈ ℂ → ((𝑥 − 1) + 1) = 𝑥)
1614, 15syl 17 . . . . . . . 8 (𝑥 ∈ (1..^𝑁) → ((𝑥 − 1) + 1) = 𝑥)
1716eqcomd 2612 . . . . . . 7 (𝑥 ∈ (1..^𝑁) → 𝑥 = ((𝑥 − 1) + 1))
1817adantl 480 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) → 𝑥 = ((𝑥 − 1) + 1))
199, 12, 18rspcedvd 3285 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ (1..^𝑁)) → ∃𝑦 ∈ (0..^(𝑁 − 1))𝑥 = (𝑦 + 1))
20 fveq2 6085 . . . . . . . 8 (𝑥 = (𝑦 + 1) → ((𝐹 cyclShift 𝐽)‘𝑥) = ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)))
21203ad2ant3 1076 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → ((𝐹 cyclShift 𝐽)‘𝑥) = ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)))
22 elfzoelz 12291 . . . . . . . . . . . . . 14 (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ ℤ)
2322zcnd 11312 . . . . . . . . . . . . 13 (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ ℂ)
2423adantl 480 . . . . . . . . . . . 12 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝑦 ∈ ℂ)
25 elfzoelz 12291 . . . . . . . . . . . . . . 15 (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
2625zcnd 11312 . . . . . . . . . . . . . 14 (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℂ)
27263ad2ant3 1076 . . . . . . . . . . . . 13 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝐽 ∈ ℂ)
2827adantr 479 . . . . . . . . . . . 12 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝐽 ∈ ℂ)
29 1cnd 9909 . . . . . . . . . . . 12 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 1 ∈ ℂ)
30 add32r 10103 . . . . . . . . . . . 12 ((𝑦 ∈ ℂ ∧ 𝐽 ∈ ℂ ∧ 1 ∈ ℂ) → (𝑦 + (𝐽 + 1)) = ((𝑦 + 1) + 𝐽))
3124, 28, 29, 30syl3anc 1317 . . . . . . . . . . 11 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + (𝐽 + 1)) = ((𝑦 + 1) + 𝐽))
3231oveq1d 6539 . . . . . . . . . 10 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝑦 + (𝐽 + 1)) mod (#‘𝐹)) = (((𝑦 + 1) + 𝐽) mod (#‘𝐹)))
3332fveq2d 6089 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝐹‘((𝑦 + (𝐽 + 1)) mod (#‘𝐹))) = (𝐹‘(((𝑦 + 1) + 𝐽) mod (#‘𝐹))))
34 simpl1 1056 . . . . . . . . . 10 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝐹 ∈ Word 𝑆)
3525peano2zd 11314 . . . . . . . . . . . 12 (𝐽 ∈ (0..^𝑁) → (𝐽 + 1) ∈ ℤ)
36353ad2ant3 1076 . . . . . . . . . . 11 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐽 + 1) ∈ ℤ)
3736adantr 479 . . . . . . . . . 10 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝐽 + 1) ∈ ℤ)
38 fzossrbm1 12318 . . . . . . . . . . . . . . 15 (𝑁 ∈ ℤ → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
392, 38syl 17 . . . . . . . . . . . . . 14 (𝐽 ∈ (0..^𝑁) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
4039sseld 3563 . . . . . . . . . . . . 13 (𝐽 ∈ (0..^𝑁) → (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ (0..^𝑁)))
41403ad2ant3 1076 . . . . . . . . . . . 12 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (0..^(𝑁 − 1)) → 𝑦 ∈ (0..^𝑁)))
4241imp 443 . . . . . . . . . . 11 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝑦 ∈ (0..^𝑁))
43 oveq2 6532 . . . . . . . . . . . . . 14 (𝑁 = (#‘𝐹) → (0..^𝑁) = (0..^(#‘𝐹)))
4443eleq2d 2669 . . . . . . . . . . . . 13 (𝑁 = (#‘𝐹) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘𝐹))))
45443ad2ant2 1075 . . . . . . . . . . . 12 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘𝐹))))
4645adantr 479 . . . . . . . . . . 11 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 ∈ (0..^𝑁) ↔ 𝑦 ∈ (0..^(#‘𝐹))))
4742, 46mpbid 220 . . . . . . . . . 10 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝑦 ∈ (0..^(#‘𝐹)))
48 cshwidxmod 13343 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift (𝐽 + 1))‘𝑦) = (𝐹‘((𝑦 + (𝐽 + 1)) mod (#‘𝐹))))
4934, 37, 47, 48syl3anc 1317 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝐹 cyclShift (𝐽 + 1))‘𝑦) = (𝐹‘((𝑦 + (𝐽 + 1)) mod (#‘𝐹))))
50253ad2ant3 1076 . . . . . . . . . . 11 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝐽 ∈ ℤ)
5150adantr 479 . . . . . . . . . 10 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → 𝐽 ∈ ℤ)
52 fzo0ss1 12319 . . . . . . . . . . . 12 (1..^𝑁) ⊆ (0..^𝑁)
5323ad2ant3 1076 . . . . . . . . . . . . 13 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝑁 ∈ ℤ)
5453, 3sylan 486 . . . . . . . . . . . 12 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (1..^𝑁))
5552, 54sseldi 3562 . . . . . . . . . . 11 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (0..^𝑁))
5643eleq2d 2669 . . . . . . . . . . . . 13 (𝑁 = (#‘𝐹) → ((𝑦 + 1) ∈ (0..^𝑁) ↔ (𝑦 + 1) ∈ (0..^(#‘𝐹))))
57563ad2ant2 1075 . . . . . . . . . . . 12 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝑦 + 1) ∈ (0..^𝑁) ↔ (𝑦 + 1) ∈ (0..^(#‘𝐹))))
5857adantr 479 . . . . . . . . . . 11 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝑦 + 1) ∈ (0..^𝑁) ↔ (𝑦 + 1) ∈ (0..^(#‘𝐹))))
5955, 58mpbid 220 . . . . . . . . . 10 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → (𝑦 + 1) ∈ (0..^(#‘𝐹)))
60 cshwidxmod 13343 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ ∧ (𝑦 + 1) ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = (𝐹‘(((𝑦 + 1) + 𝐽) mod (#‘𝐹))))
6134, 51, 59, 60syl3anc 1317 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = (𝐹‘(((𝑦 + 1) + 𝐽) mod (#‘𝐹))))
6233, 49, 613eqtr4rd 2651 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1))) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = ((𝐹 cyclShift (𝐽 + 1))‘𝑦))
63623adant3 1073 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → ((𝐹 cyclShift 𝐽)‘(𝑦 + 1)) = ((𝐹 cyclShift (𝐽 + 1))‘𝑦))
6421, 63eqtrd 2640 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → ((𝐹 cyclShift 𝐽)‘𝑥) = ((𝐹 cyclShift (𝐽 + 1))‘𝑦))
6564eqeq1d 2608 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (0..^(𝑁 − 1)) ∧ 𝑥 = (𝑦 + 1)) → (((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧 ↔ ((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧))
667, 19, 65rexxfrd2 4803 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧 ↔ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧))
6766abbidv 2724 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧} = {𝑧 ∣ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧})
6825anim2i 590 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆𝐽 ∈ ℤ))
69683adant2 1072 . . . . . 6 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆𝐽 ∈ ℤ))
70 cshwfn 13341 . . . . . 6 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ) → (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)))
7169, 70syl 17 . . . . 5 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)))
72 fnfun 5885 . . . . . . 7 ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → Fun (𝐹 cyclShift 𝐽))
7372adantl 480 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → Fun (𝐹 cyclShift 𝐽))
74433ad2ant2 1075 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^𝑁) = (0..^(#‘𝐹)))
7552, 74syl5sseq 3612 . . . . . . . 8 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
7675adantr 479 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
77 fndm 5887 . . . . . . . 8 ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
7877adantl 480 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
7976, 78sseqtr4d 3601 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽))
8073, 79jca 552 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
8171, 80mpdan 698 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
82 dfimafn 6137 . . . 4 ((Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = {𝑧 ∣ ∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧})
8381, 82syl 17 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = {𝑧 ∣ ∃𝑥 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑥) = 𝑧})
8435anim2i 590 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ))
85843adant2 1072 . . . . . 6 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ))
86 cshwfn 13341 . . . . . 6 ((𝐹 ∈ Word 𝑆 ∧ (𝐽 + 1) ∈ ℤ) → (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)))
8785, 86syl 17 . . . . 5 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)))
88 fnfun 5885 . . . . . . 7 ((𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)) → Fun (𝐹 cyclShift (𝐽 + 1)))
8988adantl 480 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → Fun (𝐹 cyclShift (𝐽 + 1)))
90393ad2ant3 1076 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^(𝑁 − 1)) ⊆ (0..^𝑁))
91 oveq2 6532 . . . . . . . . . . 11 ((#‘𝐹) = 𝑁 → (0..^(#‘𝐹)) = (0..^𝑁))
9291eqcoms 2614 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → (0..^(#‘𝐹)) = (0..^𝑁))
93923ad2ant2 1075 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^(#‘𝐹)) = (0..^𝑁))
9490, 93sseqtr4d 3601 . . . . . . . 8 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (0..^(𝑁 − 1)) ⊆ (0..^(#‘𝐹)))
9594adantr 479 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → (0..^(𝑁 − 1)) ⊆ (0..^(#‘𝐹)))
96 fndm 5887 . . . . . . . 8 ((𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹)) → dom (𝐹 cyclShift (𝐽 + 1)) = (0..^(#‘𝐹)))
9796adantl 480 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → dom (𝐹 cyclShift (𝐽 + 1)) = (0..^(#‘𝐹)))
9895, 97sseqtr4d 3601 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1)))
9989, 98jca 552 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift (𝐽 + 1)) Fn (0..^(#‘𝐹))) → (Fun (𝐹 cyclShift (𝐽 + 1)) ∧ (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1))))
10087, 99mpdan 698 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun (𝐹 cyclShift (𝐽 + 1)) ∧ (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1))))
101 dfimafn 6137 . . . 4 ((Fun (𝐹 cyclShift (𝐽 + 1)) ∧ (0..^(𝑁 − 1)) ⊆ dom (𝐹 cyclShift (𝐽 + 1))) → ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))) = {𝑧 ∣ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧})
102100, 101syl 17 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))) = {𝑧 ∣ ∃𝑦 ∈ (0..^(𝑁 − 1))((𝐹 cyclShift (𝐽 + 1))‘𝑦) = 𝑧})
10367, 83, 1023eqtr4d 2650 . 2 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
1041, 103eqtrd 2640 1 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift (𝐽 + 1)) “ (0..^(𝑁 − 1))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2592  wrex 2893  cdif 3533  wss 3536  {csn 4121  dom cdm 5025  cima 5028  Fun wfun 5781   Fn wfn 5782  cfv 5787  (class class class)co 6524  cc 9787  0cc0 9789  1c1 9790   + caddc 9792  cmin 10114  cz 11207  ..^cfzo 12286   mod cmo 12482  #chash 12931  Word cword 13089   cyclShift ccsh 13328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866  ax-pre-sup 9867
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 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-sup 8205  df-inf 8206  df-card 8622  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-div 10531  df-nn 10865  df-2 10923  df-n0 11137  df-z 11208  df-uz 11517  df-rp 11662  df-ico 12005  df-fz 12150  df-fzo 12287  df-fl 12407  df-mod 12483  df-hash 12932  df-word 13097  df-concat 13099  df-substr 13101  df-csh 13329
This theorem is referenced by:  eucrct2eupth  41412
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