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Theorem pthdlem1 29923

Description: Lemma 1 for pthd 29926. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.)

**Hypotheses**
Ref	Expression
pthd.p	⊢ (𝜑 → 𝑃 ∈ Word V)
pthd.r	⊢ 𝑅 = ((♯‘𝑃) − 1)
pthd.s	⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))

**Assertion**
Ref	Expression
pthdlem1	⊢ (𝜑 → Fun ^◡(𝑃 ↾ (1..^𝑅)))

Distinct variable groups: 𝑃,𝑖,𝑗 𝑅,𝑖,𝑗 𝜑,𝑖,𝑗

**Proof of Theorem pthdlem1**
Step	Hyp	Ref	Expression
1		pthd.p	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ Word V)
2		wrdf 14525	. . . . . . . 8 ⊢ (𝑃 ∈ Word V → 𝑃:(0..^(♯‘𝑃))⟶V)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃:(0..^(♯‘𝑃))⟶V)
4		fzo0ss1 13689	. . . . . . . . 9 ⊢ (1..^𝑅) ⊆ (0..^𝑅)
5		pthd.r	. . . . . . . . . . 11 ⊢ 𝑅 = ((♯‘𝑃) − 1)
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 = ((♯‘𝑃) − 1))
7	6	oveq2d 7407	. . . . . . . . 9 ⊢ (𝜑 → (0..^𝑅) = (0..^((♯‘𝑃) − 1)))
8	4, 7	sseqtrid 3976	. . . . . . . 8 ⊢ (𝜑 → (1..^𝑅) ⊆ (0..^((♯‘𝑃) − 1)))
9		lencl 14540	. . . . . . . . 9 ⊢ (𝑃 ∈ Word V → (♯‘𝑃) ∈ ℕ₀)
10		nn0z 12586	. . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℤ)
11		fzossrbm1 13688	. . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℤ → (0..^((♯‘𝑃) − 1)) ⊆ (0..^(♯‘𝑃)))
12	1, 9, 10, 11	4syl 19	. . . . . . . 8 ⊢ (𝜑 → (0..^((♯‘𝑃) − 1)) ⊆ (0..^(♯‘𝑃)))
13	8, 12	sstrd 3944	. . . . . . 7 ⊢ (𝜑 → (1..^𝑅) ⊆ (0..^(♯‘𝑃)))
14	3, 13	fssresd 6726	. . . . . 6 ⊢ (𝜑 → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V)
15	14	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V)
16		pthd.s	. . . . . . 7 ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))
17	16	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))
18	1, 9	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑃) ∈ ℕ₀)
19		nn0re 12484	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℝ)
20	19	ltm1d 12118	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((♯‘𝑃) − 1) < (♯‘𝑃))
21		1re 11175	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
22		peano2rem 11492	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑃) ∈ ℝ → ((♯‘𝑃) − 1) ∈ ℝ)
23	19, 22	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((♯‘𝑃) − 1) ∈ ℝ)
24		lttr 11253	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ ((♯‘𝑃) − 1) ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 < (♯‘𝑃)))
25	21, 23, 19, 24	mp3an2i 1486	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 < (♯‘𝑃)))
26		1red 11176	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) ∈ ℕ₀ → 1 ∈ ℝ)
27		ltle 11265	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → (1 < (♯‘𝑃) → 1 ≤ (♯‘𝑃)))
28	26, 19, 27	syl2anc 593	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < (♯‘𝑃) → 1 ≤ (♯‘𝑃)))
29	25, 28	syld 47	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 ≤ (♯‘𝑃)))
30	20, 29	mpan2d 704	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) → 1 ≤ (♯‘𝑃)))
31	30	imdistani 576	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → ((♯‘𝑃) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑃)))
32		elnnnn0c 12520	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℕ ↔ ((♯‘𝑃) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑃)))
33	31, 32	sylibr 236	. . . . . . . . . . . 12 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℕ)
34	18, 33	sylan 589	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℕ)
35		fzo0sn0fzo1 13755	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) ∈ ℕ → (0..^(♯‘𝑃)) = ({0} ∪ (1..^(♯‘𝑃))))
36	34, 35	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (0..^(♯‘𝑃)) = ({0} ∪ (1..^(♯‘𝑃))))
37		1zzd 12596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → 1 ∈ ℤ)
38		1p1e2 12335	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
39		2z 12597	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	38, 39	eqeltri 2857	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) ∈ ℤ
41	40	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (1 + 1) ∈ ℤ)
42	10	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℤ)
43		ltaddsub 11655	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 + 1) < (♯‘𝑃) ↔ 1 < ((♯‘𝑃) − 1)))
44	43	bicomd 225	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → (1 < ((♯‘𝑃) − 1) ↔ (1 + 1) < (♯‘𝑃)))
45	21, 26, 19, 44	mp3an2i 1486	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) ↔ (1 + 1) < (♯‘𝑃)))
46		2re 12286	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
47	38, 46	eqeltri 2857	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) ∈ ℝ
48		ltle 11265	. . . . . . . . . . . . . . . . 17 ⊢ (((1 + 1) ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 + 1) < (♯‘𝑃) → (1 + 1) ≤ (♯‘𝑃)))
49	47, 19, 48	sylancr 596	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 + 1) < (♯‘𝑃) → (1 + 1) ≤ (♯‘𝑃)))
50	45, 49	sylbid 242	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) → (1 + 1) ≤ (♯‘𝑃)))
51	50	imp 410	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (1 + 1) ≤ (♯‘𝑃))
52		eluz2 12839	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)) ↔ ((1 + 1) ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ ∧ (1 + 1) ≤ (♯‘𝑃)))
53	41, 42, 51, 52	syl3anbrc 1356	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)))
54	18, 53	sylan 589	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)))
55		fzosplitsnm1 13740	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1))) → (1..^(♯‘𝑃)) = ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))
56	37, 54, 55	syl2anc 593	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (1..^(♯‘𝑃)) = ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))
57	56	uneq2d 4119	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ({0} ∪ (1..^(♯‘𝑃))) = ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})))
58	36, 57	eqtrd 2796	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (0..^(♯‘𝑃)) = ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})))
59	58	raleqdv 3319	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ ∀𝑖 ∈ ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))))
60		ralunb 4147	. . . . . . . . 9 ⊢ (∀𝑖 ∈ ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ (∀𝑖 ∈ {0}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))))
61		ralunb 4147	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ {((♯‘𝑃) − 1)}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))))
62	61	anbi2i 632	. . . . . . . . 9 ⊢ ((∀𝑖 ∈ {0}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) ↔ (∀𝑖 ∈ {0}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ {((♯‘𝑃) − 1)}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))))
63	60, 62	bitri 277	. . . . . . . 8 ⊢ (∀𝑖 ∈ ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ (∀𝑖 ∈ {0}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ {((♯‘𝑃) − 1)}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))))
64	59, 63	bitrdi 289	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ (∀𝑖 ∈ {0}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ {((♯‘𝑃) − 1)}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))))))
65	5	eqcomi 2770	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) − 1) = 𝑅
66	65	oveq2i 7402	. . . . . . . . . . 11 ⊢ (1..^((♯‘𝑃) − 1)) = (1..^𝑅)
67	66	raleqi 3317	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ↔ ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))
68		fvres 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1..^𝑅) → ((𝑃 ↾ (1..^𝑅))‘𝑖) = (𝑃‘𝑖))
69	68	eqcomd 2767	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1..^𝑅) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
70	69	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
71	70	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
72		fvres 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1..^𝑅) → ((𝑃 ↾ (1..^𝑅))‘𝑗) = (𝑃‘𝑗))
73	72	eqcomd 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1..^𝑅) → (𝑃‘𝑗) = ((𝑃 ↾ (1..^𝑅))‘𝑗))
74	73	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → (𝑃‘𝑗) = ((𝑃 ↾ (1..^𝑅))‘𝑗))
75	71, 74	neeq12d 3017	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → ((𝑃‘𝑖) ≠ (𝑃‘𝑗) ↔ ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗)))
76	75	biimpd 231	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → ((𝑃‘𝑖) ≠ (𝑃‘𝑗) → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗)))
77	76	imim2d 57	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → ((𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) → (𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
78	77	ralimdva 3173	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) → (∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) → ∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
79	78	ralimdva 3173	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
80	67, 79	biimtrid 244	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
81	80	adantrd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ((∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ {((♯‘𝑃) − 1)}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗))) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
82	81	adantld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ((∀𝑖 ∈ {0}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ (∀𝑖 ∈ (1..^((♯‘𝑃) − 1))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) ∧ ∀𝑖 ∈ {((♯‘𝑃) − 1)}∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
83	64, 82	sylbid 242	. . . . . 6 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
84	17, 83	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗)))
85		dff14a 7249	. . . . 5 ⊢ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V ↔ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗))))
86	15, 84, 85	sylanbrc 592	. . . 4 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V)
87		df-f1 6521	. . . 4 ⊢ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V ↔ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ Fun ^◡(𝑃 ↾ (1..^𝑅))))
88	86, 87	sylib 220	. . 3 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ Fun ^◡(𝑃 ↾ (1..^𝑅))))
89	88	simprd 499	. 2 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → Fun ^◡(𝑃 ↾ (1..^𝑅)))
90		funcnv0 6582	. . 3 ⊢ Fun ^◡∅
91	18	nn0zd 12587	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑃) ∈ ℤ)
92		peano2zm 12608	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℤ → ((♯‘𝑃) − 1) ∈ ℤ)
93	91, 92	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑃) − 1) ∈ ℤ)
94	93	zred 12671	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝑃) − 1) ∈ ℝ)
95		1red 11176	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
96	94, 95	lenltd 11323	. . . . . . . . . 10 ⊢ (𝜑 → (((♯‘𝑃) − 1) ≤ 1 ↔ ¬ 1 < ((♯‘𝑃) − 1)))
97	96	biimpar 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ((♯‘𝑃) − 1) ≤ 1)
98	5, 97	eqbrtrid 5132	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → 𝑅 ≤ 1)
99		1zzd 12596	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℤ)
100	5, 93	eqeltrid 2865	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℤ)
101	99, 100	jca 519	. . . . . . . . . 10 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑅 ∈ ℤ))
102	101	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (1 ∈ ℤ ∧ 𝑅 ∈ ℤ))
103		fzon 13680	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑅 ≤ 1 ↔ (1..^𝑅) = ∅))
104	103	bicomd 225	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑅 ∈ ℤ) → ((1..^𝑅) = ∅ ↔ 𝑅 ≤ 1))
105	102, 104	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ((1..^𝑅) = ∅ ↔ 𝑅 ≤ 1))
106	98, 105	mpbird 259	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (1..^𝑅) = ∅)
107	106	reseq2d 5961	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)) = (𝑃 ↾ ∅))
108		res0 5965	. . . . . 6 ⊢ (𝑃 ↾ ∅) = ∅
109	107, 108	eqtrdi 2812	. . . . 5 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)) = ∅)
110	109	cnveqd 5843	. . . 4 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ^◡(𝑃 ↾ (1..^𝑅)) = ^◡∅)
111	110	funeqd 6538	. . 3 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (Fun ^◡(𝑃 ↾ (1..^𝑅)) ↔ Fun ^◡∅))
112	90, 111	mpbiri 260	. 2 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → Fun ^◡(𝑃 ↾ (1..^𝑅)))
113	89, 112	pm2.61dan 822	1 ⊢ (𝜑 → Fun ^◡(𝑃 ↾ (1..^𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 Vcvv 3453 ∪ cun 3900 ⊆ wss 3902 ∅c0 4283 {csn 4579 class class class wbr 5097 ^◡ccnv 5642 ↾ cres 5645 Fun wfun 6510 ⟶wf 6512 –1-1→wf1 6513 ‘cfv 6516 (class class class)co 7391 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 < clt 11210 ≤ cle 11211 − cmin 11408 ℕcn 12204 2c2 12266 ℕ₀cn0 12475 ℤcz 12562 ℤ_≥cuz 12833 ..^cfzo 13653 ♯chash 14337 Word cword 14520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-er 8672 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-2 12274 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-fzo 13654 df-hash 14338 df-word 14521

This theorem is referenced by: pthd 29926

W3C validator