Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crctcsh1wlkn0lem7 Structured version   Visualization version   GIF version

Theorem crctcsh1wlkn0lem7 41019
 Description: Lemma for crctcsh1wlkn0 41024. (Contributed by AV, 12-Mar-2021.)
Hypotheses
Ref Expression
crctcsh1wlkn0lem.s (𝜑𝑆 ∈ (1..^𝑁))
crctcsh1wlkn0lem.q 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
crctcsh1wlkn0lem.h 𝐻 = (𝐹 cyclShift 𝑆)
crctcsh1wlkn0lem.n 𝑁 = (#‘𝐹)
crctcsh1wlkn0lem.f (𝜑𝐹 ∈ Word 𝐴)
crctcsh1wlkn0lem.p (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
crctcsh1wlkn0lem.e (𝜑 → (𝑃𝑁) = (𝑃‘0))
Assertion
Ref Expression
crctcsh1wlkn0lem7 (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
Distinct variable groups:   𝑥,𝑁   𝑥,𝑃   𝑥,𝑆   𝜑,𝑥   𝑖,𝐹   𝑖,𝐼   𝑖,𝑁   𝑃,𝑖   𝑆,𝑖   𝜑,𝑖,𝑗   𝑥,𝑗   𝑗,𝐼   𝑗,𝐻   𝑗,𝑁   𝑄,𝑗   𝑆,𝑗
Allowed substitution hints:   𝐴(𝑥,𝑖,𝑗)   𝑃(𝑗)   𝑄(𝑥,𝑖)   𝐹(𝑥,𝑗)   𝐻(𝑥,𝑖)   𝐼(𝑥)

Proof of Theorem crctcsh1wlkn0lem7
StepHypRef Expression
1 crctcsh1wlkn0lem.s . . . . . 6 (𝜑𝑆 ∈ (1..^𝑁))
2 crctcsh1wlkn0lem.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
3 crctcsh1wlkn0lem.h . . . . . 6 𝐻 = (𝐹 cyclShift 𝑆)
4 crctcsh1wlkn0lem.n . . . . . 6 𝑁 = (#‘𝐹)
5 crctcsh1wlkn0lem.f . . . . . 6 (𝜑𝐹 ∈ Word 𝐴)
6 crctcsh1wlkn0lem.p . . . . . 6 (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
71, 2, 3, 4, 5, 6crctcsh1wlkn0lem4 41016 . . . . 5 (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
8 eqidd 2611 . . . . . . 7 (𝜑 → (𝑁𝑆) = (𝑁𝑆))
9 crctcsh1wlkn0lem.e . . . . . . . 8 (𝜑 → (𝑃𝑁) = (𝑃‘0))
101, 2, 3, 4, 5, 6, 9crctcsh1wlkn0lem6 41018 . . . . . . 7 ((𝜑 ∧ (𝑁𝑆) = (𝑁𝑆)) → if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
118, 10mpdan 699 . . . . . 6 (𝜑 → if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
12 ovex 6577 . . . . . . 7 (𝑁𝑆) ∈ V
13 1wlkslem1 40809 . . . . . . 7 (𝑗 = (𝑁𝑆) → (if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆))))))
1412, 13ralsn 4169 . . . . . 6 (∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
1511, 14sylibr 223 . . . . 5 (𝜑 → ∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
16 ralunb 3756 . . . . 5 (∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ (∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ∧ ∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
177, 15, 16sylanbrc 695 . . . 4 (𝜑 → ∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
18 elfzo1 12385 . . . . . . 7 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
19 nnz 11276 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
20 nnz 11276 . . . . . . . . . . . 12 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
21 zsubcl 11296 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
2219, 20, 21syl2anr 494 . . . . . . . . . . 11 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
23223adant3 1074 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℤ)
24 nnre 10904 . . . . . . . . . . . 12 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
25 nnre 10904 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
26 posdif 10400 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 ↔ 0 < (𝑁𝑆)))
27 0re 9919 . . . . . . . . . . . . . 14 0 ∈ ℝ
28 resubcl 10224 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
2928ancoms 468 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
30 ltle 10005 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑁𝑆) ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3127, 29, 30sylancr 694 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3226, 31sylbid 229 . . . . . . . . . . . 12 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
3324, 25, 32syl2an 493 . . . . . . . . . . 11 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
34333impia 1253 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 0 ≤ (𝑁𝑆))
35 elnn0z 11267 . . . . . . . . . 10 ((𝑁𝑆) ∈ ℕ0 ↔ ((𝑁𝑆) ∈ ℤ ∧ 0 ≤ (𝑁𝑆)))
3623, 34, 35sylanbrc 695 . . . . . . . . 9 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ0)
37 elnn0uz 11601 . . . . . . . . 9 ((𝑁𝑆) ∈ ℕ0 ↔ (𝑁𝑆) ∈ (ℤ‘0))
3836, 37sylib 207 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ (ℤ‘0))
39 fzosplitsn 12442 . . . . . . . 8 ((𝑁𝑆) ∈ (ℤ‘0) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4038, 39syl 17 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4118, 40sylbi 206 . . . . . 6 (𝑆 ∈ (1..^𝑁) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
421, 41syl 17 . . . . 5 (𝜑 → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4342raleqdv 3121 . . . 4 (𝜑 → (∀𝑗 ∈ (0..^((𝑁𝑆) + 1))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ ∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
4417, 43mpbird 246 . . 3 (𝜑 → ∀𝑗 ∈ (0..^((𝑁𝑆) + 1))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
451, 2, 3, 4, 5, 6crctcsh1wlkn0lem5 41017 . . 3 (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
46 ralunb 3756 . . 3 (∀𝑗 ∈ ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ (∀𝑗 ∈ (0..^((𝑁𝑆) + 1))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ∧ ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
4744, 45, 46sylanbrc 695 . 2 (𝜑 → ∀𝑗 ∈ ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
48 nnsub 10936 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 ↔ (𝑁𝑆) ∈ ℕ))
4948biimp3a 1424 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ)
50 nnnn0 11176 . . . . . . 7 ((𝑁𝑆) ∈ ℕ → (𝑁𝑆) ∈ ℕ0)
51 peano2nn0 11210 . . . . . . 7 ((𝑁𝑆) ∈ ℕ0 → ((𝑁𝑆) + 1) ∈ ℕ0)
5249, 50, 513syl 18 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ ℕ0)
53 nnnn0 11176 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
54533ad2ant2 1076 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℕ0)
5525anim1i 590 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
5655ancoms 468 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
57 crctcsh1wlkn0lem1 41013 . . . . . . . 8 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
5856, 57syl 17 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
59583adant3 1074 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ≤ 𝑁)
60 elfz2nn0 12300 . . . . . 6 (((𝑁𝑆) + 1) ∈ (0...𝑁) ↔ (((𝑁𝑆) + 1) ∈ ℕ0𝑁 ∈ ℕ0 ∧ ((𝑁𝑆) + 1) ≤ 𝑁))
6152, 54, 59, 60syl3anbrc 1239 . . . . 5 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
6218, 61sylbi 206 . . . 4 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
63 fzosplit 12370 . . . 4 (((𝑁𝑆) + 1) ∈ (0...𝑁) → (0..^𝑁) = ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁)))
641, 62, 633syl 18 . . 3 (𝜑 → (0..^𝑁) = ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁)))
6564raleqdv 3121 . 2 (𝜑 → (∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))) ↔ ∀𝑗 ∈ ((0..^((𝑁𝑆) + 1)) ∪ (((𝑁𝑆) + 1)..^𝑁))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗)))))
6647, 65mpbird 246 1 (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  if-wif 1006   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∪ cun 3538   ⊆ wss 3540  ifcif 4036  {csn 4125  {cpr 4127   class class class wbr 4583   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   cyclShift ccsh 13385 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386 This theorem is referenced by:  crctcsh1wlkn0  41024
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