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Theorem pthdlem1 29834

Description: Lemma 1 for pthd 29837. (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 14480	. . . . . . . 8 ⊢ (𝑃 ∈ Word V → 𝑃:(0..^(♯‘𝑃))⟶V)
3	1, 2	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑃:(0..^(♯‘𝑃))⟶V)
4		fzo0ss1 13644	. . . . . . . . 9 ⊢ (1..^𝑅) ⊆ (0..^𝑅)
5		pthd.r	. . . . . . . . . . 11 ⊢ 𝑅 = ((♯‘𝑃) − 1)
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 = ((♯‘𝑃) − 1))
7	6	oveq2d 7383	. . . . . . . . 9 ⊢ (𝜑 → (0..^𝑅) = (0..^((♯‘𝑃) − 1)))
8	4, 7	sseqtrid 3964	. . . . . . . 8 ⊢ (𝜑 → (1..^𝑅) ⊆ (0..^((♯‘𝑃) − 1)))
9		lencl 14495	. . . . . . . . 9 ⊢ (𝑃 ∈ Word V → (♯‘𝑃) ∈ ℕ₀)
10		nn0z 12548	. . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℤ)
11		fzossrbm1 13643	. . . . . . . . 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 3932	. . . . . . 7 ⊢ (𝜑 → (1..^𝑅) ⊆ (0..^(♯‘𝑃)))
14	3, 13	fssresd 6707	. . . . . 6 ⊢ (𝜑 → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V)
15	14	adantr 480	. . . . 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 480	. . . . . 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 12446	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℝ)
20	19	ltm1d 12088	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((♯‘𝑃) − 1) < (♯‘𝑃))
21		1re 11144	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
22		peano2rem 11461	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑃) ∈ ℝ → ((♯‘𝑃) − 1) ∈ ℝ)
23	19, 22	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((♯‘𝑃) − 1) ∈ ℝ)
24		lttr 11222	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ ((♯‘𝑃) − 1) ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 < (♯‘𝑃)))
25	21, 23, 19, 24	mp3an2i 1469	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 < (♯‘𝑃)))
26		1red 11145	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) ∈ ℕ₀ → 1 ∈ ℝ)
27		ltle 11234	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → (1 < (♯‘𝑃) → 1 ≤ (♯‘𝑃)))
28	26, 19, 27	syl2anc 585	. . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) → 1 ≤ (♯‘𝑃)))
31	30	imdistani 568	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → ((♯‘𝑃) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑃)))
32		elnnnn0c 12482	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℕ ↔ ((♯‘𝑃) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑃)))
33	31, 32	sylibr 234	. . . . . . . . . . . 12 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℕ)
34	18, 33	sylan 581	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℕ)
35		fzo0sn0fzo1 13710	. . . . . . . . . . 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 12558	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → 1 ∈ ℤ)
38		1p1e2 12301	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
39		2z 12559	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	38, 39	eqeltri 2832	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) ∈ ℤ
41	40	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (1 + 1) ∈ ℤ)
42	10	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℤ)
43		ltaddsub 11624	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 + 1) < (♯‘𝑃) ↔ 1 < ((♯‘𝑃) − 1)))
44	43	bicomd 223	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → (1 < ((♯‘𝑃) − 1) ↔ (1 + 1) < (♯‘𝑃)))
45	21, 26, 19, 44	mp3an2i 1469	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) ↔ (1 + 1) < (♯‘𝑃)))
46		2re 12255	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
47	38, 46	eqeltri 2832	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) ∈ ℝ
48		ltle 11234	. . . . . . . . . . . . . . . . 17 ⊢ (((1 + 1) ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 + 1) < (♯‘𝑃) → (1 + 1) ≤ (♯‘𝑃)))
49	47, 19, 48	sylancr 588	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 + 1) < (♯‘𝑃) → (1 + 1) ≤ (♯‘𝑃)))
50	45, 49	sylbid 240	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) → (1 + 1) ≤ (♯‘𝑃)))
51	50	imp 406	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (1 + 1) ≤ (♯‘𝑃))
52		eluz2 12794	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)) ↔ ((1 + 1) ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ ∧ (1 + 1) ≤ (♯‘𝑃)))
53	41, 42, 51, 52	syl3anbrc 1345	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)))
54	18, 53	sylan 581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)))
55		fzosplitsnm1 13695	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1))) → (1..^(♯‘𝑃)) = ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))
56	37, 54, 55	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (1..^(♯‘𝑃)) = ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))
57	56	uneq2d 4108	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ({0} ∪ (1..^(♯‘𝑃))) = ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})))
58	36, 57	eqtrd 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (0..^(♯‘𝑃)) = ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})))
59	58	raleqdv 3295	. . . . . . . 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 4137	. . . . . . . . 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 4137	. . . . . . . . . 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 624	. . . . . . . . 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 275	. . . . . . . 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 287	. . . . . . 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 2745	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) − 1) = 𝑅
66	65	oveq2i 7378	. . . . . . . . . . 11 ⊢ (1..^((♯‘𝑃) − 1)) = (1..^𝑅)
67	66	raleqi 3293	. . . . . . . . . 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 6859	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1..^𝑅) → ((𝑃 ↾ (1..^𝑅))‘𝑖) = (𝑃‘𝑖))
69	68	eqcomd 2742	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1..^𝑅) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
70	69	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
71	70	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
72		fvres 6859	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1..^𝑅) → ((𝑃 ↾ (1..^𝑅))‘𝑗) = (𝑃‘𝑗))
73	72	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1..^𝑅) → (𝑃‘𝑗) = ((𝑃 ↾ (1..^𝑅))‘𝑗))
74	73	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → (𝑃‘𝑗) = ((𝑃 ↾ (1..^𝑅))‘𝑗))
75	71, 74	neeq12d 2993	. . . . . . . . . . . . . 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 229	. . . . . . . . . . . . 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 3149	. . . . . . . . . . 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 3149	. . . . . . . . . 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 242	. . . . . . . . 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 491	. . . . . . . 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 490	. . . . . . 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 240	. . . . . 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 7225	. . . . 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 584	. . . 4 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V)
87		df-f1 6503	. . . 4 ⊢ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V ↔ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ Fun ^◡(𝑃 ↾ (1..^𝑅))))
88	86, 87	sylib 218	. . 3 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ Fun ^◡(𝑃 ↾ (1..^𝑅))))
89	88	simprd 495	. 2 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → Fun ^◡(𝑃 ↾ (1..^𝑅)))
90		funcnv0 6564	. . 3 ⊢ Fun ^◡∅
91	18	nn0zd 12549	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑃) ∈ ℤ)
92		peano2zm 12570	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℤ → ((♯‘𝑃) − 1) ∈ ℤ)
93	91, 92	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑃) − 1) ∈ ℤ)
94	93	zred 12633	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝑃) − 1) ∈ ℝ)
95		1red 11145	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
96	94, 95	lenltd 11292	. . . . . . . . . 10 ⊢ (𝜑 → (((♯‘𝑃) − 1) ≤ 1 ↔ ¬ 1 < ((♯‘𝑃) − 1)))
97	96	biimpar 477	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ((♯‘𝑃) − 1) ≤ 1)
98	5, 97	eqbrtrid 5120	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → 𝑅 ≤ 1)
99		1zzd 12558	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℤ)
100	5, 93	eqeltrid 2840	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℤ)
101	99, 100	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑅 ∈ ℤ))
102	101	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (1 ∈ ℤ ∧ 𝑅 ∈ ℤ))
103		fzon 13635	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑅 ≤ 1 ↔ (1..^𝑅) = ∅))
104	103	bicomd 223	. . . . . . . . 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 257	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (1..^𝑅) = ∅)
107	106	reseq2d 5944	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)) = (𝑃 ↾ ∅))
108		res0 5948	. . . . . 6 ⊢ (𝑃 ↾ ∅) = ∅
109	107, 108	eqtrdi 2787	. . . . 5 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)) = ∅)
110	109	cnveqd 5830	. . . 4 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ^◡(𝑃 ↾ (1..^𝑅)) = ^◡∅)
111	110	funeqd 6520	. . 3 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (Fun ^◡(𝑃 ↾ (1..^𝑅)) ↔ Fun ^◡∅))
112	90, 111	mpbiri 258	. 2 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → Fun ^◡(𝑃 ↾ (1..^𝑅)))
113	89, 112	pm2.61dan 813	1 ⊢ (𝜑 → Fun ^◡(𝑃 ↾ (1..^𝑅)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 Vcvv 3429 ∪ cun 3887 ⊆ wss 3889 ∅c0 4273 {csn 4567 class class class wbr 5085 ^◡ccnv 5630 ↾ cres 5633 Fun wfun 6492 ⟶wf 6494 –1-1→wf1 6495 ‘cfv 6498 (class class class)co 7367 ℝcr 11037 0cc0 11038 1c1 11039 + caddc 11041 < clt 11179 ≤ cle 11180 − cmin 11377 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ..^cfzo 13608 ♯chash 14292 Word cword 14475

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-fzo 13609 df-hash 14293 df-word 14476

This theorem is referenced by: pthd 29837

W3C validator