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Theorem cshf1 14858

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 6817	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → 𝐹:(0..^(♯‘𝐹))⟶𝐴)
2		iswrdi 14566	. . . . 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 14848	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
5	4	3adant1 1130	. . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
6	5	adantr 480	. . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
7		feq1 6728	. . . . . . . 8 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴))
8	7	adantl 481	. . . . . . 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 257	. . . . . 6 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))⟶𝐴)
10		dff13 7292	. . . . . . . 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 6919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
12	11	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
13	12	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
14		cshwidxmod 14851	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝑖 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))))
15	14	3expia 1121	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹)))))
16	15	3adant1 1130	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 6919	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
22	21	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
23	22	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
24		cshwidxmod 14851	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
25	24	3expia 1121	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹)))))
26	25	3adant1 1130	. . . . . . . . . . . . . . . . . 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 490	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 2780	. . . . . . . . . . . . . . 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 2756	. . . . . . . . . . . . . 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 714	. . . . . . . . . . . . 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 13757	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) ↔ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑖 < (♯‘𝐹)))
33		nn0z 12664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℤ)
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → 𝑖 ∈ ℤ)
35	34	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑖 ∈ ℤ)
36		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
37	35, 36	zaddcld 12751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (𝑖 + 𝑆) ∈ ℤ)
38		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (♯‘𝐹) ∈ ℕ)
39	38	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (♯‘𝐹) ∈ ℕ)
40	37, 39	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
41	40	ex 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑆 ∈ ℤ → ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
42	41	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
46	45	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
47	46	impcom 407	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
48		zmodfzo 13945	. . . . . . . . . . . . . . . . . 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 13757	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0..^(♯‘𝐹)) ↔ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑗 < (♯‘𝐹)))
51		nn0z 12664	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
52	51	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → 𝑗 ∈ ℤ)
53	52	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑗 ∈ ℤ)
54		simpl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
55	53, 54	zaddcld 12751	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (𝑗 + 𝑆) ∈ ℤ)
56		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (♯‘𝐹) ∈ ℕ)
57	56	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (♯‘𝐹) ∈ ℕ)
58	55, 57	jca 511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
59	58	expcom 413	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
60	59	3adant3 1132	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑗 < (♯‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
61	50, 60	sylbi 217	. . . . . . . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(♯‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
64	63	adantld 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
65	64	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
66		zmodfzo 13945	. . . . . . . . . . . . . . . . . 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 6929	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (♯‘𝐹)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘𝑦)))
69		eqeq1 2744	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (♯‘𝐹)) → (𝑥 = 𝑦 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = 𝑦))
70	68, 69	imbi12d 344	. . . . . . . . . . . . . . . . . 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 6920	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → (𝐹‘𝑦) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
72	71	eqeq2d 2751	. . . . . . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = 𝑦 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
74	72, 73	imbi12d 344	. . . . . . . . . . . . . . . . . 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 3646	. . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 13997	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
79	78	3expa 1118	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
80	79	ancoms 458	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
81	80	bicomd 223	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹))))
82	81	3ad2antl3 1187	. . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 240	. . . . . . . . . . . 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 3208	. . . . . . . . . . 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 1352	. . . . . . . . . 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 481	. . . . . . . 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 217	. . . . . . 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 1347	. . . . . 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 511	. . . . 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 1352	. . . 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 1111	. 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 7292	. 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 234	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 class class class wbr 5166 ⟶wf 6569 –1-1→wf1 6570 ‘cfv 6573 (class class class)co 7448 0cc0 11184 + caddc 11187 < clt 11324 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ..^cfzo 13711 mod cmo 13920 ♯chash 14379 Word cword 14562 cyclShift ccsh 14836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-inf 9512 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-n0 12554 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-hash 14380 df-word 14563 df-concat 14619 df-substr 14689 df-pfx 14719 df-csh 14837

This theorem is referenced by: cshinj 14859 cshf1o 32929

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