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Theorem cshimadifsn 13368
Description: The image of a cyclically shifted word under its domain without its left 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
cshimadifsn ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))

Proof of Theorem cshimadifsn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrdfn 13116 . . . . . 6 (𝐹 ∈ Word 𝑆𝐹 Fn (0..^(#‘𝐹)))
2 fnfun 5884 . . . . . 6 (𝐹 Fn (0..^(#‘𝐹)) → Fun 𝐹)
31, 2syl 17 . . . . 5 (𝐹 ∈ Word 𝑆 → Fun 𝐹)
433ad2ant1 1074 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → Fun 𝐹)
5 wrddm 13109 . . . . . 6 (𝐹 ∈ Word 𝑆 → dom 𝐹 = (0..^(#‘𝐹)))
6 difssd 3695 . . . . . . . . 9 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → ((0..^(#‘𝐹)) ∖ {𝐽}) ⊆ (0..^(#‘𝐹)))
7 oveq2 6531 . . . . . . . . . . 11 (𝑁 = (#‘𝐹) → (0..^𝑁) = (0..^(#‘𝐹)))
87difeq1d 3684 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → ((0..^𝑁) ∖ {𝐽}) = ((0..^(#‘𝐹)) ∖ {𝐽}))
98adantl 480 . . . . . . . . 9 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → ((0..^𝑁) ∖ {𝐽}) = ((0..^(#‘𝐹)) ∖ {𝐽}))
10 simpl 471 . . . . . . . . 9 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → dom 𝐹 = (0..^(#‘𝐹)))
116, 9, 103sstr4d 3606 . . . . . . . 8 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹)
1211a1d 25 . . . . . . 7 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → (𝐽 ∈ (0..^𝑁) → ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹))
1312ex 448 . . . . . 6 (dom 𝐹 = (0..^(#‘𝐹)) → (𝑁 = (#‘𝐹) → (𝐽 ∈ (0..^𝑁) → ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹)))
145, 13syl 17 . . . . 5 (𝐹 ∈ Word 𝑆 → (𝑁 = (#‘𝐹) → (𝐽 ∈ (0..^𝑁) → ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹)))
15143imp 1248 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹)
164, 15jca 552 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun 𝐹 ∧ ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹))
17 dfimafn 6136 . . 3 ((Fun 𝐹 ∧ ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = {𝑧 ∣ ∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧})
1816, 17syl 17 . 2 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = {𝑧 ∣ ∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧})
19 modsumfzodifsn 12556 . . . . . . 7 ((𝐽 ∈ (0..^𝑁) ∧ 𝑦 ∈ (1..^𝑁)) → ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
20193ad2antl3 1217 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
21 oveq2 6531 . . . . . . . . . 10 ((#‘𝐹) = 𝑁 → ((𝑦 + 𝐽) mod (#‘𝐹)) = ((𝑦 + 𝐽) mod 𝑁))
2221eqcoms 2613 . . . . . . . . 9 (𝑁 = (#‘𝐹) → ((𝑦 + 𝐽) mod (#‘𝐹)) = ((𝑦 + 𝐽) mod 𝑁))
2322eleq1d 2667 . . . . . . . 8 (𝑁 = (#‘𝐹) → (((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}) ↔ ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽})))
24233ad2ant2 1075 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}) ↔ ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽})))
2524adantr 479 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → (((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}) ↔ ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽})))
2620, 25mpbird 245 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → ((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}))
27 modfzo0difsn 12555 . . . . . . 7 ((𝐽 ∈ (0..^𝑁) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁))
28273ad2antl3 1217 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁))
29 oveq2 6531 . . . . . . . . . . 11 (𝑁 = (#‘𝐹) → ((𝑦 + 𝐽) mod 𝑁) = ((𝑦 + 𝐽) mod (#‘𝐹)))
3029eqcomd 2611 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → ((𝑦 + 𝐽) mod (#‘𝐹)) = ((𝑦 + 𝐽) mod 𝑁))
3130eqeq2d 2615 . . . . . . . . 9 (𝑁 = (#‘𝐹) → (𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) ↔ 𝑥 = ((𝑦 + 𝐽) mod 𝑁)))
3231rexbidv 3029 . . . . . . . 8 (𝑁 = (#‘𝐹) → (∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) ↔ ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁)))
33323ad2ant2 1075 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) ↔ ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁)))
3433adantr 479 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → (∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) ↔ ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁)))
3528, 34mpbird 245 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)))
36 fveq2 6084 . . . . . . . 8 (𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) → (𝐹𝑥) = (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))))
37363ad2ant3 1076 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁) ∧ 𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹))) → (𝐹𝑥) = (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))))
38 simpl1 1056 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → 𝐹 ∈ Word 𝑆)
39 elfzoelz 12290 . . . . . . . . . . 11 (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
40393ad2ant3 1076 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝐽 ∈ ℤ)
4140adantr 479 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → 𝐽 ∈ ℤ)
42 oveq2 6531 . . . . . . . . . . . . 13 (𝑁 = (#‘𝐹) → (1..^𝑁) = (1..^(#‘𝐹)))
4342eleq2d 2668 . . . . . . . . . . . 12 (𝑁 = (#‘𝐹) → (𝑦 ∈ (1..^𝑁) ↔ 𝑦 ∈ (1..^(#‘𝐹))))
44 fzo0ss1 12318 . . . . . . . . . . . . 13 (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
4544sseli 3559 . . . . . . . . . . . 12 (𝑦 ∈ (1..^(#‘𝐹)) → 𝑦 ∈ (0..^(#‘𝐹)))
4643, 45syl6bi 241 . . . . . . . . . . 11 (𝑁 = (#‘𝐹) → (𝑦 ∈ (1..^𝑁) → 𝑦 ∈ (0..^(#‘𝐹))))
47463ad2ant2 1075 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (1..^𝑁) → 𝑦 ∈ (0..^(#‘𝐹))))
4847imp 443 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → 𝑦 ∈ (0..^(#‘𝐹)))
49 cshwidxmod 13342 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝐽)‘𝑦) = (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))))
5049eqcomd 2611 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))) = ((𝐹 cyclShift 𝐽)‘𝑦))
5138, 41, 48, 50syl3anc 1317 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))) = ((𝐹 cyclShift 𝐽)‘𝑦))
52513adant3 1073 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁) ∧ 𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹))) → (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))) = ((𝐹 cyclShift 𝐽)‘𝑦))
5337, 52eqtrd 2639 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁) ∧ 𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹))) → (𝐹𝑥) = ((𝐹 cyclShift 𝐽)‘𝑦))
5453eqeq1d 2607 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁) ∧ 𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹))) → ((𝐹𝑥) = 𝑧 ↔ ((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧))
5526, 35, 54rexxfrd2 4802 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧 ↔ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧))
5655abbidv 2723 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧} = {𝑧 ∣ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧})
5739anim2i 590 . . . . . . . 8 ((𝐹 ∈ Word 𝑆𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆𝐽 ∈ ℤ))
58573adant2 1072 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆𝐽 ∈ ℤ))
59 cshwfn 13340 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ) → (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)))
6058, 59syl 17 . . . . . 6 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)))
61 fnfun 5884 . . . . . . . 8 ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → Fun (𝐹 cyclShift 𝐽))
6261adantl 480 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → Fun (𝐹 cyclShift 𝐽))
6342, 44syl6eqss 3613 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
64633ad2ant2 1075 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
6564adantr 479 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
66 fndm 5886 . . . . . . . . 9 ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
6766adantl 480 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
6865, 67sseqtr4d 3600 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽))
6962, 68jca 552 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
7060, 69mpdan 698 . . . . 5 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
71 dfimafn 6136 . . . . 5 ((Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = {𝑧 ∣ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧})
7270, 71syl 17 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((𝐹 cyclShift 𝐽) “ (1..^𝑁)) = {𝑧 ∣ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧})
7372eqcomd 2611 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧} = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
7456, 73eqtrd 2639 . 2 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧} = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
7518, 74eqtrd 2639 1 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 “ ((0..^𝑁) ∖ {𝐽})) = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1975  {cab 2591  wrex 2892  cdif 3532  wss 3535  {csn 4120  dom cdm 5024  cima 5027  Fun wfun 5780   Fn wfn 5781  cfv 5786  (class class class)co 6523  0cc0 9788  1c1 9789   + caddc 9791  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-ico 12004  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:  cshimadifsn0  13369
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