cshf1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cshf1		Structured version Visualization version GIF version

Theorem cshf1 14698

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 6738	. . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→𝐴 → 𝐹:(0..^(♯‘𝐹))⟶𝐴)
2		iswrdi 14406	. . . . 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 14688	. . . . . . . . 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 481	. . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴)
7		feq1 6649	. . . . . . . 8 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺:(0..^(♯‘𝐹))⟶𝐴 ↔ (𝐹 cyclShift 𝑆):(0..^(♯‘𝐹))⟶𝐴))
8	7	adantl 482	. . . . . . 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 256	. . . . . 6 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(♯‘𝐹))⟶𝐴)
10		dff13 7202	. . . . . . . 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 6841	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
14		cshwidxmod 14691	. . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 6841	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
24		cshwidxmod 14691	. . . . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . 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 407	. . . . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . . . 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 2752	. . . . . . . . . . . . . 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 13613	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) ↔ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑖 < (♯‘𝐹)))
33		nn0z 12524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℤ)
34	33	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → 𝑖 ∈ ℤ)
35	34	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑖 ∈ ℤ)
36		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
37	35, 36	zaddcld 12611	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (𝑖 + 𝑆) ∈ ℤ)
38		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (♯‘𝐹) ∈ ℕ)
39	38	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (♯‘𝐹) ∈ ℕ)
40	37, 39	jca 512	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
41	40	ex 413	. . . . . . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (0..^(♯‘𝐹)) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
46	45	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
47	46	impcom 408	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑖 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
48		zmodfzo 13799	. . . . . . . . . . . . . . . . . 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 13613	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0..^(♯‘𝐹)) ↔ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ ∧ 𝑗 < (♯‘𝐹)))
51		nn0z 12524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑗 ∈ ℕ₀ → 𝑗 ∈ ℤ)
52	51	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → 𝑗 ∈ ℤ)
53	52	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑗 ∈ ℤ)
54		simpl 483	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
55	53, 54	zaddcld 12611	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (𝑗 + 𝑆) ∈ ℤ)
56		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ) → (♯‘𝐹) ∈ ℕ)
57	56	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → (♯‘𝐹) ∈ ℕ)
58	55, 57	jca 512	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (♯‘𝐹) ∈ ℕ)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
59	58	expcom 414	. . . . . . . . . . . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ)))
65	64	imp 407	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (♯‘𝐹) ∈ ℕ))
66		zmodfzo 13799	. . . . . . . . . . . . . . . . . 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 6851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (♯‘𝐹)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (♯‘𝐹))) = (𝐹‘𝑦)))
69		eqeq1 2740	. . . . . . . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (♯‘𝐹)) → (𝐹‘𝑦) = (𝐹‘((𝑗 + 𝑆) mod (♯‘𝐹))))
72	71	eqeq2d 2747	. . . . . . . . . . . . . . . . . . 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 2748	. . . . . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 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 13851	. . . . . . . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(♯‘𝐹)) ∧ 𝑗 ∈ (0..^(♯‘𝐹)))) → (((𝑖 + 𝑆) mod (♯‘𝐹)) = ((𝑗 + 𝑆) mod (♯‘𝐹)) ↔ 𝑖 = 𝑗))
81	80	bicomd 222	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . 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 452	. . . . . . . . . . . . . 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 407	. . . . . . . . . . . . 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 239	. . . . . . . . . . . 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 3197	. . . . . . . . . . 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 482	. . . . . . . 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 216	. . . . . . 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 512	. . . . 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 7202	. 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 233	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 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 class class class wbr 5105 ⟶wf 6492 –1-1→wf1 6493 ‘cfv 6496 (class class class)co 7357 0cc0 11051 + caddc 11054 < clt 11189 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ..^cfzo 13567 mod cmo 13774 ♯chash 14230 Word cword 14402 cyclShift ccsh 14676

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-rp 12916 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-hash 14231 df-word 14403 df-concat 14459 df-substr 14529 df-pfx 14559 df-csh 14677

This theorem is referenced by: cshinj 14699 cshf1o 31816

W3C validator