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Theorem crctcshwlkn0lem7 26702
Description: Lemma for crctcshwlkn0 26707. (Contributed by AV, 12-Mar-2021.)
Hypotheses
Ref Expression
crctcshwlkn0lem.s (𝜑𝑆 ∈ (1..^𝑁))
crctcshwlkn0lem.q 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
crctcshwlkn0lem.h 𝐻 = (𝐹 cyclShift 𝑆)
crctcshwlkn0lem.n 𝑁 = (#‘𝐹)
crctcshwlkn0lem.f (𝜑𝐹 ∈ Word 𝐴)
crctcshwlkn0lem.p (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))
crctcshwlkn0lem.e (𝜑 → (𝑃𝑁) = (𝑃‘0))
Assertion
Ref Expression
crctcshwlkn0lem7 (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 crctcshwlkn0lem7
StepHypRef Expression
1 crctcshwlkn0lem.s . . . . . 6 (𝜑𝑆 ∈ (1..^𝑁))
2 crctcshwlkn0lem.q . . . . . 6 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
3 crctcshwlkn0lem.h . . . . . 6 𝐻 = (𝐹 cyclShift 𝑆)
4 crctcshwlkn0lem.n . . . . . 6 𝑁 = (#‘𝐹)
5 crctcshwlkn0lem.f . . . . . 6 (𝜑𝐹 ∈ Word 𝐴)
6 crctcshwlkn0lem.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, 6crctcshwlkn0lem4 26699 . . . . 5 (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
8 eqidd 2622 . . . . . . 7 (𝜑 → (𝑁𝑆) = (𝑁𝑆))
9 crctcshwlkn0lem.e . . . . . . . 8 (𝜑 → (𝑃𝑁) = (𝑃‘0))
101, 2, 3, 4, 5, 6, 9crctcshwlkn0lem6 26701 . . . . . . 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 702 . . . . . 6 (𝜑 → if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
12 ovex 6675 . . . . . . 7 (𝑁𝑆) ∈ V
13 wkslem1 26497 . . . . . . 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 4220 . . . . . 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 224 . . . . 5 (𝜑 → ∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
16 ralunb 3792 . . . . 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 698 . . . 4 (𝜑 → ∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
18 elfzo1 12513 . . . . . . 7 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
19 nnz 11396 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
20 nnz 11396 . . . . . . . . . . . 12 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
21 zsubcl 11416 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
2219, 20, 21syl2anr 495 . . . . . . . . . . 11 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
23223adant3 1080 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℤ)
24 nnre 11024 . . . . . . . . . . . 12 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
25 nnre 11024 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
26 posdif 10518 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 ↔ 0 < (𝑁𝑆)))
27 0re 10037 . . . . . . . . . . . . . 14 0 ∈ ℝ
28 resubcl 10342 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
2928ancoms 469 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
30 ltle 10123 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑁𝑆) ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3127, 29, 30sylancr 695 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3226, 31sylbid 230 . . . . . . . . . . . 12 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
3324, 25, 32syl2an 494 . . . . . . . . . . 11 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
34333impia 1260 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 0 ≤ (𝑁𝑆))
35 elnn0z 11387 . . . . . . . . . 10 ((𝑁𝑆) ∈ ℕ0 ↔ ((𝑁𝑆) ∈ ℤ ∧ 0 ≤ (𝑁𝑆)))
3623, 34, 35sylanbrc 698 . . . . . . . . 9 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ0)
37 elnn0uz 11722 . . . . . . . . 9 ((𝑁𝑆) ∈ ℕ0 ↔ (𝑁𝑆) ∈ (ℤ‘0))
3836, 37sylib 208 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ (ℤ‘0))
39 fzosplitsn 12572 . . . . . . . 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 207 . . . . . 6 (𝑆 ∈ (1..^𝑁) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
421, 41syl 17 . . . . 5 (𝜑 → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4342raleqdv 3142 . . . 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 247 . . 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, 6crctcshwlkn0lem5 26700 . . 3 (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
46 ralunb 3792 . . 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 698 . 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 11056 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 ↔ (𝑁𝑆) ∈ ℕ))
4948biimp3a 1431 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ)
50 nnnn0 11296 . . . . . . 7 ((𝑁𝑆) ∈ ℕ → (𝑁𝑆) ∈ ℕ0)
51 peano2nn0 11330 . . . . . . 7 ((𝑁𝑆) ∈ ℕ0 → ((𝑁𝑆) + 1) ∈ ℕ0)
5249, 50, 513syl 18 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ ℕ0)
53 nnnn0 11296 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
54533ad2ant2 1082 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℕ0)
5525anim1i 592 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
5655ancoms 469 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
57 crctcshwlkn0lem1 26696 . . . . . . . 8 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
5856, 57syl 17 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
59583adant3 1080 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ≤ 𝑁)
60 elfz2nn0 12427 . . . . . 6 (((𝑁𝑆) + 1) ∈ (0...𝑁) ↔ (((𝑁𝑆) + 1) ∈ ℕ0𝑁 ∈ ℕ0 ∧ ((𝑁𝑆) + 1) ≤ 𝑁))
6152, 54, 59, 60syl3anbrc 1245 . . . . 5 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
6218, 61sylbi 207 . . . 4 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
63 fzosplit 12497 . . . 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 3142 . 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 247 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 384  if-wif 1012  w3a 1037   = wceq 1482  wcel 1989  wral 2911  cun 3570  wss 3572  ifcif 4084  {csn 4175  {cpr 4177   class class class wbr 4651  cmpt 4727  cfv 5886  (class class class)co 6647  cr 9932  0cc0 9933  1c1 9934   + caddc 9936   < clt 10071  cle 10072  cmin 10263  cn 11017  0cn0 11289  cz 11374  cuz 11684  ...cfz 12323  ..^cfzo 12461  #chash 13112  Word cword 13286   cyclShift ccsh 13528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-cnex 9989  ax-resscn 9990  ax-1cn 9991  ax-icn 9992  ax-addcl 9993  ax-addrcl 9994  ax-mulcl 9995  ax-mulrcl 9996  ax-mulcom 9997  ax-addass 9998  ax-mulass 9999  ax-distr 10000  ax-i2m1 10001  ax-1ne0 10002  ax-1rid 10003  ax-rnegex 10004  ax-rrecex 10005  ax-cnre 10006  ax-pre-lttri 10007  ax-pre-lttrn 10008  ax-pre-ltadd 10009  ax-pre-mulgt0 10010  ax-pre-sup 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-nel 2897  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-int 4474  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-om 7063  df-1st 7165  df-2nd 7166  df-wrecs 7404  df-recs 7465  df-rdg 7503  df-1o 7557  df-oadd 7561  df-er 7739  df-en 7953  df-dom 7954  df-sdom 7955  df-fin 7956  df-sup 8345  df-inf 8346  df-card 8762  df-pnf 10073  df-mnf 10074  df-xr 10075  df-ltxr 10076  df-le 10077  df-sub 10265  df-neg 10266  df-div 10682  df-nn 11018  df-2 11076  df-n0 11290  df-z 11375  df-uz 11685  df-rp 11830  df-fz 12324  df-fzo 12462  df-fl 12588  df-mod 12664  df-hash 13113  df-word 13294  df-concat 13296  df-substr 13298  df-csh 13529
This theorem is referenced by:  crctcshwlkn0  26707
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