Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cshimadifsn Structured version   Visualization version   GIF version

 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)))

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wrdfn 13351 . . . . . 6 (𝐹 ∈ Word 𝑆𝐹 Fn (0..^(#‘𝐹)))
2 fnfun 6026 . . . . . 6 (𝐹 Fn (0..^(#‘𝐹)) → Fun 𝐹)
31, 2syl 17 . . . . 5 (𝐹 ∈ Word 𝑆 → Fun 𝐹)
433ad2ant1 1102 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → Fun 𝐹)
5 wrddm 13344 . . . . . 6 (𝐹 ∈ Word 𝑆 → dom 𝐹 = (0..^(#‘𝐹)))
6 difssd 3771 . . . . . . . . 9 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → ((0..^(#‘𝐹)) ∖ {𝐽}) ⊆ (0..^(#‘𝐹)))
7 oveq2 6698 . . . . . . . . . . 11 (𝑁 = (#‘𝐹) → (0..^𝑁) = (0..^(#‘𝐹)))
87difeq1d 3760 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → ((0..^𝑁) ∖ {𝐽}) = ((0..^(#‘𝐹)) ∖ {𝐽}))
98adantl 481 . . . . . . . . 9 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → ((0..^𝑁) ∖ {𝐽}) = ((0..^(#‘𝐹)) ∖ {𝐽}))
10 simpl 472 . . . . . . . . 9 ((dom 𝐹 = (0..^(#‘𝐹)) ∧ 𝑁 = (#‘𝐹)) → dom 𝐹 = (0..^(#‘𝐹)))
116, 9, 103sstr4d 3681 . . . . . . . 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 449 . . . . . 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 1275 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹)
164, 15jca 553 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun 𝐹 ∧ ((0..^𝑁) ∖ {𝐽}) ⊆ dom 𝐹))
17 dfimafn 6284 . . 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 12783 . . . . . . 7 ((𝐽 ∈ (0..^𝑁) ∧ 𝑦 ∈ (1..^𝑁)) → ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
20193ad2antl3 1245 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽}))
21 oveq2 6698 . . . . . . . . . 10 ((#‘𝐹) = 𝑁 → ((𝑦 + 𝐽) mod (#‘𝐹)) = ((𝑦 + 𝐽) mod 𝑁))
2221eqcoms 2659 . . . . . . . . 9 (𝑁 = (#‘𝐹) → ((𝑦 + 𝐽) mod (#‘𝐹)) = ((𝑦 + 𝐽) mod 𝑁))
2322eleq1d 2715 . . . . . . . 8 (𝑁 = (#‘𝐹) → (((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}) ↔ ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽})))
24233ad2ant2 1103 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}) ↔ ((𝑦 + 𝐽) mod 𝑁) ∈ ((0..^𝑁) ∖ {𝐽})))
2524adantr 480 . . . . . 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 247 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → ((𝑦 + 𝐽) mod (#‘𝐹)) ∈ ((0..^𝑁) ∖ {𝐽}))
27 modfzo0difsn 12782 . . . . . . 7 ((𝐽 ∈ (0..^𝑁) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁))
28273ad2antl3 1245 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁))
29 oveq2 6698 . . . . . . . . . . 11 (𝑁 = (#‘𝐹) → ((𝑦 + 𝐽) mod 𝑁) = ((𝑦 + 𝐽) mod (#‘𝐹)))
3029eqcomd 2657 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → ((𝑦 + 𝐽) mod (#‘𝐹)) = ((𝑦 + 𝐽) mod 𝑁))
3130eqeq2d 2661 . . . . . . . . 9 (𝑁 = (#‘𝐹) → (𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) ↔ 𝑥 = ((𝑦 + 𝐽) mod 𝑁)))
3231rexbidv 3081 . . . . . . . 8 (𝑁 = (#‘𝐹) → (∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) ↔ ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod 𝑁)))
33323ad2ant2 1103 . . . . . . 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 480 . . . . . 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 247 . . . . 5 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑥 ∈ ((0..^𝑁) ∖ {𝐽})) → ∃𝑦 ∈ (1..^𝑁)𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)))
36 fveq2 6229 . . . . . . . 8 (𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹)) → (𝐹𝑥) = (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))))
37363ad2ant3 1104 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁) ∧ 𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹))) → (𝐹𝑥) = (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))))
38 simpl1 1084 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → 𝐹 ∈ Word 𝑆)
39 elfzoelz 12509 . . . . . . . . . . 11 (𝐽 ∈ (0..^𝑁) → 𝐽 ∈ ℤ)
40393ad2ant3 1104 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → 𝐽 ∈ ℤ)
4140adantr 480 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → 𝐽 ∈ ℤ)
42 oveq2 6698 . . . . . . . . . . . . 13 (𝑁 = (#‘𝐹) → (1..^𝑁) = (1..^(#‘𝐹)))
4342eleq2d 2716 . . . . . . . . . . . 12 (𝑁 = (#‘𝐹) → (𝑦 ∈ (1..^𝑁) ↔ 𝑦 ∈ (1..^(#‘𝐹))))
44 fzo0ss1 12537 . . . . . . . . . . . . 13 (1..^(#‘𝐹)) ⊆ (0..^(#‘𝐹))
4544sseli 3632 . . . . . . . . . . . 12 (𝑦 ∈ (1..^(#‘𝐹)) → 𝑦 ∈ (0..^(#‘𝐹)))
4643, 45syl6bi 243 . . . . . . . . . . 11 (𝑁 = (#‘𝐹) → (𝑦 ∈ (1..^𝑁) → 𝑦 ∈ (0..^(#‘𝐹))))
47463ad2ant2 1103 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝑦 ∈ (1..^𝑁) → 𝑦 ∈ (0..^(#‘𝐹))))
4847imp 444 . . . . . . . . 9 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → 𝑦 ∈ (0..^(#‘𝐹)))
49 cshwidxmod 13595 . . . . . . . . . 10 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝐽)‘𝑦) = (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))))
5049eqcomd 2657 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝐽 ∈ ℤ ∧ 𝑦 ∈ (0..^(#‘𝐹))) → (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))) = ((𝐹 cyclShift 𝐽)‘𝑦))
5138, 41, 48, 50syl3anc 1366 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁)) → (𝐹‘((𝑦 + 𝐽) mod (#‘𝐹))) = ((𝐹 cyclShift 𝐽)‘𝑦))
52513adant3 1101 . . . . . . 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 2685 . . . . . 6 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ 𝑦 ∈ (1..^𝑁) ∧ 𝑥 = ((𝑦 + 𝐽) mod (#‘𝐹))) → (𝐹𝑥) = ((𝐹 cyclShift 𝐽)‘𝑦))
5453eqeq1d 2653 . . . . 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 4915 . . . 4 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧 ↔ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧))
5655abbidv 2770 . . 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 592 . . . . . . . 8 ((𝐹 ∈ Word 𝑆𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆𝐽 ∈ ℤ))
58573adant2 1100 . . . . . . 7 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (𝐹 ∈ Word 𝑆𝐽 ∈ ℤ))
59 cshwfn 13593 . . . . . . 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 6026 . . . . . . . 8 ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → Fun (𝐹 cyclShift 𝐽))
6261adantl 481 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → Fun (𝐹 cyclShift 𝐽))
6342, 44syl6eqss 3688 . . . . . . . . . 10 (𝑁 = (#‘𝐹) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
64633ad2ant2 1103 . . . . . . . . 9 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
6564adantr 480 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ (0..^(#‘𝐹)))
66 fndm 6028 . . . . . . . . 9 ((𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹)) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
6766adantl 481 . . . . . . . 8 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → dom (𝐹 cyclShift 𝐽) = (0..^(#‘𝐹)))
6865, 67sseqtr4d 3675 . . . . . . 7 (((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) ∧ (𝐹 cyclShift 𝐽) Fn (0..^(#‘𝐹))) → (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽))
6962, 68jca 553 . . . . . 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 703 . . . . 5 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → (Fun (𝐹 cyclShift 𝐽) ∧ (1..^𝑁) ⊆ dom (𝐹 cyclShift 𝐽)))
71 dfimafn 6284 . . . . 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 2657 . . 3 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑦 ∈ (1..^𝑁)((𝐹 cyclShift 𝐽)‘𝑦) = 𝑧} = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
7456, 73eqtrd 2685 . 2 ((𝐹 ∈ Word 𝑆𝑁 = (#‘𝐹) ∧ 𝐽 ∈ (0..^𝑁)) → {𝑧 ∣ ∃𝑥 ∈ ((0..^𝑁) ∖ {𝐽})(𝐹𝑥) = 𝑧} = ((𝐹 cyclShift 𝐽) “ (1..^𝑁)))
7518, 74eqtrd 2685 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942   ∖ cdif 3604   ⊆ wss 3607  {csn 4210  dom cdm 5143   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  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-ico 12219  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:  cshimadifsn0  13622
 Copyright terms: Public domain W3C validator