cshf1 - Metamath Proof Explorer

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Theorem cshf1 13407

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 6014	. . . . 5 ⊢ (𝐹:(0..^(#‘𝐹))–1-1→𝐴 → 𝐹:(0..^(#‘𝐹))⟶𝐴)
2		iswrdi 13164	. . . . 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 13397	. . . . . . . . 9 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝐹 cyclShift 𝑆):(0..^(#‘𝐹))⟶𝐴)
5	4	3adant1 1072	. . . . . . . 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 5939	. . . . . . . 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 246	. . . . . 6 ⊢ (((𝐹:(0..^(#‘𝐹))–1-1→𝐴 ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ 𝐺 = (𝐹 cyclShift 𝑆)) → 𝐺:(0..^(#‘𝐹))⟶𝐴)
10		dff13 6416	. . . . . . . 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 6102	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺‘𝑖) = ((𝐹 cyclShift 𝑆)‘𝑖))
12	11	3ad2ant1 1075	. . . . . . . . . . . . . . . . 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 13400	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝑖 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
15	14	3expia 1259	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑖 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑖) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹)))))
16	15	3adant1 1072	. . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 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 6102	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐺 = (𝐹 cyclShift 𝑆) → (𝐺‘𝑗) = ((𝐹 cyclShift 𝑆)‘𝑗))
22	21	3ad2ant1 1075	. . . . . . . . . . . . . . . . 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 13400	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
25	24	3expia 1259	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝐹 cyclShift 𝑆)‘𝑗) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
26	25	3adant1 1072	. . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . 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 444	. . . . . . . . . . . . . . . 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 2644	. . . . . . . . . . . . . . 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 2625	. . . . . . . . . . . . . 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 747	. . . . . . . . . . . . 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 12376	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (0..^(#‘𝐹)) ↔ (𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ ∧ 𝑖 < (#‘𝐹)))
33		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
37	35, 36	zaddcld 11362	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → (𝑖 + 𝑆) ∈ ℤ)
38		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ) → (#‘𝐹) ∈ ℕ)
39	38	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → (#‘𝐹) ∈ ℕ)
40	37, 39	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
41	40	ex 449	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑆 ∈ ℤ → ((𝑖 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
42	41	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . . . . 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 1074	. . . . . . . . . . . . . . . . . . . . 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 206	. . . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑖 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
48		zmodfzo 12555	. . . . . . . . . . . . . . . . . 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 12376	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑗 ∈ (0..^(#‘𝐹)) ↔ (𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ ∧ 𝑗 < (#‘𝐹)))
51		nn0z 11277	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → 𝑆 ∈ ℤ)
55	53, 54	zaddcld 11362	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → (𝑗 + 𝑆) ∈ ℤ)
56		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ) → (#‘𝐹) ∈ ℕ)
57	56	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → (#‘𝐹) ∈ ℕ)
58	55, 57	jca 553	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑆 ∈ ℤ ∧ (𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
59	58	expcom 450	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
60	59	3adant3 1074	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ ℕ₀ ∧ (#‘𝐹) ∈ ℕ ∧ 𝑗 < (#‘𝐹)) → (𝑆 ∈ ℤ → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
61	50, 60	sylbi 206	. . . . . . . . . . . . . . . . . . . . . 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 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → (𝑗 ∈ (0..^(#‘𝐹)) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
64	63	adantld 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ)))
65	64	imp 444	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺 = (𝐹 cyclShift 𝑆) ∧ 𝐹 ∈ Word 𝐴 ∧ 𝑆 ∈ ℤ) ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → ((𝑗 + 𝑆) ∈ ℤ ∧ (#‘𝐹) ∈ ℕ))
66		zmodfzo 12555	. . . . . . . . . . . . . . . . . 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		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (𝐹‘𝑥) = (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))))
69	68	eqeq1d 2612	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘𝑦)))
70		eqeq1 2614	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (𝑥 = 𝑦 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦))
71	69, 70	imbi12d 333	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ((𝑖 + 𝑆) mod (#‘𝐹)) → (((𝐹‘𝑥) = (𝐹‘𝑦) → 𝑥 = 𝑦) ↔ ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘𝑦) → ((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦)))
72		fveq2 6103	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (𝐹‘𝑦) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹))))
73	72	eqeq2d 2620	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → ((𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘𝑦) ↔ (𝐹‘((𝑖 + 𝑆) mod (#‘𝐹))) = (𝐹‘((𝑗 + 𝑆) mod (#‘𝐹)))))
74		eqeq2 2621	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = ((𝑗 + 𝑆) mod (#‘𝐹)) → (((𝑖 + 𝑆) mod (#‘𝐹)) = 𝑦 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
75	73, 74	imbi12d 333	. . . . . . . . . . . . . . . . . 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)))))
76	71, 75	rspc2v 3293	. . . . . . . . . . . . . . . . 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)))))
77	49, 67, 76	syl2anc 691	. . . . . . . . . . . . . . . 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)))))
78		simpr 476	. . . . . . . . . . . . . . . . . 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))))
79		addmodlteq 12607	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
80	79	3expa 1257	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) ∧ 𝑆 ∈ ℤ) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
81	80	ancoms 468	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)) ↔ 𝑖 = 𝑗))
82	81	bicomd 212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑆 ∈ ℤ ∧ (𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹)))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹))))
83	82	ex 449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑆 ∈ ℤ → ((𝑖 ∈ (0..^(#‘𝐹)) ∧ 𝑗 ∈ (0..^(#‘𝐹))) → (𝑖 = 𝑗 ↔ ((𝑖 + 𝑆) mod (#‘𝐹)) = ((𝑗 + 𝑆) mod (#‘𝐹)))))
84	83	3ad2ant3 1077	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((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)))))
85	84	imp 444	. . . . . . . . . . . . . . . . . . 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))))
86	85	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))))
87	78, 86	sylibrd 248	. . . . . . . . . . . . . . . . 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))
88	87	ex 449	. . . . . . . . . . . . . . . 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)))
89	77, 88	syld 46	. . . . . . . . . . . . . . 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)))
90	89	impancom 455	. . . . . . . . . . . . . 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)))
91	90	imp 444	. . . . . . . . . . . . 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))
92	31, 91	sylbid 229	. . . . . . . . . . . 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))
93	92	ralrimivva 2954	. . . . . . . . . . 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))
94	93	3exp1 1275	. . . . . . . . . 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)))))
95	94	com14 94	. . . . . . . . 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)))))
96	95	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)))))
97	10, 96	sylbi 206	. . . . . . 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)))))
98	97	3imp1 1272	. . . . . 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))
99	9, 98	jca 553	. . . . 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)))
100	99	3exp1 1275	. . . 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))))))
101	3, 100	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)))))
102	101	3imp 1249	. 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)))
103		dff13 6416	. 2 ⊢ (𝐺:(0..^(#‘𝐹))–1-1→𝐴 ↔ (𝐺:(0..^(#‘𝐹))⟶𝐴 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))∀𝑗 ∈ (0..^(#‘𝐹))((𝐺‘𝑖) = (𝐺‘𝑗) → 𝑖 = 𝑗)))
104	102, 103	sylibr 223	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 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 class class class wbr 4583 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 (class class class)co 6549 0cc0 9815 + caddc 9818 < clt 9953 ℕcn 10897 ℕ₀cn0 11169 ℤcz 11254 ..^cfzo 12334 mod cmo 12530 #chash 12979 Word cword 13146 cyclShift ccsh 13385

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-hash 12980 df-word 13154 df-concat 13156 df-substr 13158 df-csh 13386

This theorem is referenced by: cshinj 13408

W3C validator