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Theorem crctcsh1wlkn0lem7 41017
Description: Lemma for crctcsh1wlkn0 41022. (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 41014 . . . . 5 (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
8 eqidd 2606 . . . . . . 7 (𝜑 → (𝑁𝑆) = (𝑁𝑆))
9 crctcsh1wlkn0lem.e . . . . . . . 8 (𝜑 → (𝑃𝑁) = (𝑃‘0))
101, 2, 3, 4, 5, 6, 9crctcsh1wlkn0lem6 41016 . . . . . . 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 698 . . . . . 6 (𝜑 → if-((𝑄‘(𝑁𝑆)) = (𝑄‘((𝑁𝑆) + 1)), (𝐼‘(𝐻‘(𝑁𝑆))) = {(𝑄‘(𝑁𝑆))}, {(𝑄‘(𝑁𝑆)), (𝑄‘((𝑁𝑆) + 1))} ⊆ (𝐼‘(𝐻‘(𝑁𝑆)))))
12 ovex 6551 . . . . . . 7 (𝑁𝑆) ∈ V
13 1wlkslem1 40807 . . . . . . 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 4164 . . . . . 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 222 . . . . 5 (𝜑 → ∀𝑗 ∈ {(𝑁𝑆)}if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
16 ralunb 3751 . . . . 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 694 . . . 4 (𝜑 → ∀𝑗 ∈ ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)})if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
18 elfzo1 12336 . . . . . . 7 (𝑆 ∈ (1..^𝑁) ↔ (𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁))
19 nnz 11228 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℤ)
20 nnz 11228 . . . . . . . . . . . 12 (𝑆 ∈ ℕ → 𝑆 ∈ ℤ)
21 zsubcl 11248 . . . . . . . . . . . 12 ((𝑁 ∈ ℤ ∧ 𝑆 ∈ ℤ) → (𝑁𝑆) ∈ ℤ)
2219, 20, 21syl2anr 493 . . . . . . . . . . 11 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁𝑆) ∈ ℤ)
23223adant3 1073 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℤ)
24 nnre 10870 . . . . . . . . . . . 12 (𝑆 ∈ ℕ → 𝑆 ∈ ℝ)
25 nnre 10870 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → 𝑁 ∈ ℝ)
26 posdif 10366 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 ↔ 0 < (𝑁𝑆)))
27 0re 9892 . . . . . . . . . . . . . 14 0 ∈ ℝ
28 resubcl 10192 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
2928ancoms 467 . . . . . . . . . . . . . 14 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑁𝑆) ∈ ℝ)
30 ltle 9973 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ (𝑁𝑆) ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3127, 29, 30sylancr 693 . . . . . . . . . . . . 13 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (0 < (𝑁𝑆) → 0 ≤ (𝑁𝑆)))
3226, 31sylbid 228 . . . . . . . . . . . 12 ((𝑆 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
3324, 25, 32syl2an 492 . . . . . . . . . . 11 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 → 0 ≤ (𝑁𝑆)))
34333impia 1252 . . . . . . . . . 10 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 0 ≤ (𝑁𝑆))
35 elnn0z 11219 . . . . . . . . . 10 ((𝑁𝑆) ∈ ℕ0 ↔ ((𝑁𝑆) ∈ ℤ ∧ 0 ≤ (𝑁𝑆)))
3623, 34, 35sylanbrc 694 . . . . . . . . 9 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ0)
37 elnn0uz 11553 . . . . . . . . 9 ((𝑁𝑆) ∈ ℕ0 ↔ (𝑁𝑆) ∈ (ℤ‘0))
3836, 37sylib 206 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ (ℤ‘0))
39 fzosplitsn 12393 . . . . . . . 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 205 . . . . . 6 (𝑆 ∈ (1..^𝑁) → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
421, 41syl 17 . . . . 5 (𝜑 → (0..^((𝑁𝑆) + 1)) = ((0..^(𝑁𝑆)) ∪ {(𝑁𝑆)}))
4342raleqdv 3116 . . . 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 245 . . 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 41015 . . 3 (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
46 ralunb 3751 . . 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 694 . 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 10902 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑆 < 𝑁 ↔ (𝑁𝑆) ∈ ℕ))
4948biimp3a 1423 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → (𝑁𝑆) ∈ ℕ)
50 nnnn0 11142 . . . . . . 7 ((𝑁𝑆) ∈ ℕ → (𝑁𝑆) ∈ ℕ0)
51 peano2nn0 11176 . . . . . . 7 ((𝑁𝑆) ∈ ℕ0 → ((𝑁𝑆) + 1) ∈ ℕ0)
5249, 50, 513syl 18 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ ℕ0)
53 nnnn0 11142 . . . . . . 7 (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0)
54533ad2ant2 1075 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → 𝑁 ∈ ℕ0)
5525anim1i 589 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑆 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
5655ancoms 467 . . . . . . . 8 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ))
57 crctcsh1wlkn0lem1 41011 . . . . . . . 8 ((𝑁 ∈ ℝ ∧ 𝑆 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
5856, 57syl 17 . . . . . . 7 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝑁𝑆) + 1) ≤ 𝑁)
59583adant3 1073 . . . . . 6 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ≤ 𝑁)
60 elfz2nn0 12251 . . . . . 6 (((𝑁𝑆) + 1) ∈ (0...𝑁) ↔ (((𝑁𝑆) + 1) ∈ ℕ0𝑁 ∈ ℕ0 ∧ ((𝑁𝑆) + 1) ≤ 𝑁))
6152, 54, 59, 60syl3anbrc 1238 . . . . 5 ((𝑆 ∈ ℕ ∧ 𝑁 ∈ ℕ ∧ 𝑆 < 𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
6218, 61sylbi 205 . . . 4 (𝑆 ∈ (1..^𝑁) → ((𝑁𝑆) + 1) ∈ (0...𝑁))
63 fzosplit 12321 . . . 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 3116 . 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 245 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 382  if-wif 1005  w3a 1030   = wceq 1474  wcel 1975  wral 2891  cun 3533  wss 3535  ifcif 4031  {csn 4120  {cpr 4122   class class class wbr 4573  cmpt 4633  cfv 5786  (class class class)co 6523  cr 9787  0cc0 9788  1c1 9789   + caddc 9791   < clt 9926  cle 9927  cmin 10113  cn 10863  0cn0 11135  cz 11206  cuz 11515  ...cfz 12148  ..^cfzo 12285  #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-ifp 1006  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-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:  crctcsh1wlkn0  41022
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