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Theorem cshf1 14166

Description: Cyclically shifting a word which contains a symbol at most once results in a word which contains a symbol at most once. (Contributed by AV, 14-Mar-2021.)

**Assertion**
Ref	Expression
cshf1	⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))–1-1→𝐴)

Proof of Theorem cshf1 Dummy variables 𝑖 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1f 6570	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → 𝐹:(0..^(♯‘𝐹))⟶𝐴)
2		iswrdi 13859	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))⟶𝐴 → 𝐹 ∈ Word 𝐴)
3	1, 2	syl 17	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → 𝐹 ∈ Word 𝐴)
4		cshwf 14156	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
5	4	3adant1 1126	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
6	5	adantr 483	. . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
7		feq1 6490	. . . . . . . 8 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴))
8	7	adantl 484	. . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴))
9	6, 8	mpbird 259	. . . . . 6 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))⟶𝐴)
10		dff13 7007	. . . . . . . 8 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ↔ (𝐹:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)))
11		fveq1 6664	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
12	11	3ad2ant1 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
13	12	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
14		cshwidxmod 14159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))))
15	14	3expia 1117	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹)))))
16	15	3adant1 1126	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹)))))
17	16	com12 32	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹)))))
18	17	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹)))))
19	18	impcom 410	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))))
20	13, 19	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))))
21		fveq1 6664	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
22	21	3ad2ant1 1129	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
23	22	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
24		cshwidxmod 14159	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
25	24	3expia 1117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
26	25	3adant1 1126	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
27	26	adantld 493	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
28	27	imp 409	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
29	23, 28	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
30	20, 29	eqeq12d 2837	. . . . . . . . . . . . . 14 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝐺‘𝑖) = (𝐺‘𝑗) ↔ (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
31	30	adantlr 713	. . . . . . . . . . . . 13 ⊢ ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝐺‘𝑖) = (𝐺‘𝑗) ↔ (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
32		elfzo0 13072	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) ↔ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑖 < (♯‘𝐹)))
33		nn0z 11999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℤ)
34	33	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → 𝑖 ∈ ℤ)
35	34	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑖 ∈ ℤ)
36		simpl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
37	35, 36	zaddcld 12085	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (𝑖 + 𝑆) ∈ ℤ)
38		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (♯‘𝐹) ∈ ℕ)
39	38	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (♯‘𝐹) ∈ ℕ)
40	37, 39	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
41	40	ex 415	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑆 ∈ ℤ → ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
42	41	3ad2ant3 1131	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
43	42	com12 32	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
44	43	3adant3 1128	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑖 < (♯‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
45	32, 44	sylbi 219	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
46	45	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
47	46	impcom 410	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
48		zmodfzo 13256	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) mod (♯‘𝐹)) ∈ (0..^(♯‘𝐹)))
49	47, 48	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) ∈ (0..^(♯‘𝐹)))
50		elfzo0 13072	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0..^(♯‘𝐹)) ↔ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑗 < (♯‘𝐹)))
51		nn0z 11999	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
52	51	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → 𝑗 ∈ ℤ)
53	52	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑗 ∈ ℤ)
54		simpl 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
55	53, 54	zaddcld 12085	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (𝑗 + 𝑆) ∈ ℤ)
56		simpr 487	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (♯‘𝐹) ∈ ℕ)
57	56	adantl 484	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (♯‘𝐹) ∈ ℕ)
58	55, 57	jca 514	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
59	58	expcom 416	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
60	59	3adant3 1128	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑗 < (♯‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
61	50, 60	sylbi 219	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑗 ∈ (0..^(♯‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
62	61	com12 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ ℤ → (𝑗 ∈ (0..^(♯‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
63	62	3ad2ant3 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
64	63	adantld 493	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
65	64	imp 409	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
66		zmodfzo 13256	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ) → ((𝑗 + 𝑆) mod (♯‘𝐹)) ∈ (0..^(♯‘𝐹)))
67	65, 66	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑗 + 𝑆) mod (♯‘𝐹)) ∈ (0..^(♯‘𝐹)))
68		fveqeq2 6674	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (♯‘𝐹)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘𝑦)))
69		eqeq1 2825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (♯‘𝐹)) → (𝑥 = 𝑦 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = 𝑦))
70	68, 69	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (♯‘𝐹)) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘𝑦) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = 𝑦)))
71		fveq2 6665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → (𝐹‘𝑦) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
72	71	eqeq2d 2832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
73		eqeq2 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = 𝑦 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
74	72, 73	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → (((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘𝑦) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = 𝑦) ↔ ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)))))
75	70, 74	rspc2v 3633	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑖 + 𝑆) mod (♯‘𝐹)) ∈ (0..^(♯‘𝐹)) ∧ ((𝑗 + 𝑆) mod (♯‘𝐹)) ∈ (0..^(♯‘𝐹))) → (∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)))))
76	49, 67, 75	syl2anc 586	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)))))
77		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) ∧ ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)))) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
78		addmodlteq 13308	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
79	78	3expa 1114	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
80	79	ancoms 461	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
81	80	bicomd 225	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
82	81	3ad2antl3 1183	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
83	82	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) ∧ ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
84	77, 83	sylibrd 261	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) ∧ ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)))) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → 𝑖 = 𝑗))
85	84	ex 415	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → 𝑖 = 𝑗)))
86	76, 85	syld 47	. . . . . . . . . . . . . . 15 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → 𝑖 = 𝑗)))
87	86	impancom 454	. . . . . . . . . . . . . 14 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → 𝑖 = 𝑗)))
88	87	imp 409	. . . . . . . . . . . . 13 ⊢ ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))) → 𝑖 = 𝑗))
89	31, 88	sylbid 242	. . . . . . . . . . . 12 ⊢ ((((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))
90	89	ralrimivva 3191	. . . . . . . . . . 11 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))
91	90	3exp1 1348	. . . . . . . . . 10 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))))
92	91	com14 96	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))))
93	92	adantl 484	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑥 ∈ (0..^(♯‘𝐹))∀𝑦 ∈ (0..^(♯‘𝐹))((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦)) → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))))
94	10, 93	sylbi 219	. . . . . . 7 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))))
95	94	3imp1 1343	. . . . . 6 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))
96	9, 95	jca 514	. . . . 5 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))
97	96	3exp1 1348	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → (𝐹 ∈ Word 𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗))))))
98	3, 97	mpd 15	. . 3 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → (𝑆 ∈ ℤ → (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))))
99	98	3imp 1107	. 2 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))
100		dff13 7007	. 2 ⊢ (𝐺:(0..^(♯‘𝐹))–1-1→𝐴 ↔ (𝐺:(0..^(♯‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))∀𝑗 ∈ (0..^(♯‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))
101	99, 100	sylibr 236	1 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))–1-1→𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 class class class wbr 5059 ⟶wf 6346 –1-1→wf1 6347 ‘cfv 6350 (class class class)co 7150 0cc0 10531 + caddc 10534 < clt 10669 ℕcn 11632 ℕ₀cn0 11891 ℤcz 11975 ..^cfzo 13027 mod cmo 13231 ♯chash 13684 Word cword 13855 cyclShift ccsh 14144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-hash 13685 df-word 13856 df-concat 13917 df-substr 13997 df-pfx 14027 df-csh 14145

This theorem is referenced by: cshinj 14167 cshf1o 30631

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