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Theorem pthdlem1 30144

Description: Lemma 1 for pthd 30147. (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 14568	. . . . . . . 8 ⊢ (𝑃 ∈ Word V → 𝑃:(0..^(♯‘𝑃))⟶V)
3	1, 2	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑃:(0..^(♯‘𝑃))⟶V)
4		fzo0ss1 13730	. . . . . . . . 9 ⊢ (1..^𝑅) ⊆ (0..^𝑅)
5		pthd.r	. . . . . . . . . . 11 ⊢ 𝑅 = ((♯‘𝑃) − 1)
6	5	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 = ((♯‘𝑃) − 1))
7	6	oveq2d 7432	. . . . . . . . 9 ⊢ (𝜑 → (0..^𝑅) = (0..^((♯‘𝑃) − 1)))
8	4, 7	sseqtrid 3980	. . . . . . . 8 ⊢ (𝜑 → (1..^𝑅) ⊆ (0..^((♯‘𝑃) − 1)))
9		lencl 14583	. . . . . . . . 9 ⊢ (𝑃 ∈ Word V → (♯‘𝑃) ∈ ℕ₀)
10		nn0z 12626	. . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℤ)
11		fzossrbm1 13729	. . . . . . . . 9 ⊢ ((♯‘𝑃) ∈ ℤ → (0..^((♯‘𝑃) − 1)) ⊆ (0..^(♯‘𝑃)))
12	1, 9, 10, 11	4syl 20	. . . . . . . 8 ⊢ (𝜑 → (0..^((♯‘𝑃) − 1)) ⊆ (0..^(♯‘𝑃)))
13	8, 12	sstrd 3948	. . . . . . 7 ⊢ (𝜑 → (1..^𝑅) ⊆ (0..^(♯‘𝑃)))
14	3, 13	fssresd 6749	. . . . . 6 ⊢ (𝜑 → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V)
15	14	adantr 486	. . . . 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 486	. . . . . 6 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))
18	1, 9	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑃) ∈ ℕ₀)
19		nn0re 12524	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (♯‘𝑃) ∈ ℝ)
20	19	ltm1d 12158	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((♯‘𝑃) − 1) < (♯‘𝑃))
21		1re 11219	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℝ
22		peano2rem 11536	. . . . . . . . . . . . . . . . . 18 ⊢ ((♯‘𝑃) ∈ ℝ → ((♯‘𝑃) − 1) ∈ ℝ)
23	19, 22	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((♯‘𝑃) − 1) ∈ ℝ)
24		lttr 11297	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ ((♯‘𝑃) − 1) ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 < (♯‘𝑃)))
25	21, 23, 19, 24	mp3an2i 1495	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 < (♯‘𝑃)))
26		1red 11220	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑃) ∈ ℕ₀ → 1 ∈ ℝ)
27		ltle 11309	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → (1 < (♯‘𝑃) → 1 ≤ (♯‘𝑃)))
28	26, 19, 27	syl2anc 596	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < (♯‘𝑃) → 1 ≤ (♯‘𝑃)))
29	25, 28	syld 48	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 < ((♯‘𝑃) − 1) ∧ ((♯‘𝑃) − 1) < (♯‘𝑃)) → 1 ≤ (♯‘𝑃)))
30	20, 29	mpan2d 707	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) → 1 ≤ (♯‘𝑃)))
31	30	imdistani 579	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → ((♯‘𝑃) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑃)))
32		elnnnn0c 12560	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℕ ↔ ((♯‘𝑃) ∈ ℕ₀ ∧ 1 ≤ (♯‘𝑃)))
33	31, 32	sylibr 237	. . . . . . . . . . . 12 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℕ)
34	18, 33	sylan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℕ)
35		fzo0sn0fzo1 13796	. . . . . . . . . . 11 ⊢ ((♯‘𝑃) ∈ ℕ → (0..^(♯‘𝑃)) = ({0} ∪ (1..^(♯‘𝑃))))
36	34, 35	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (0..^(♯‘𝑃)) = ({0} ∪ (1..^(♯‘𝑃))))
37		1zzd 12636	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → 1 ∈ ℤ)
38		1p1e2 12375	. . . . . . . . . . . . . . . 16 ⊢ (1 + 1) = 2
39		2z 12637	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
40	38, 39	eqeltri 2861	. . . . . . . . . . . . . . 15 ⊢ (1 + 1) ∈ ℤ
41	40	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (1 + 1) ∈ ℤ)
42	10	adantr 486	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ ℤ)
43		ltaddsub 11699	. . . . . . . . . . . . . . . . . 18 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 + 1) < (♯‘𝑃) ↔ 1 < ((♯‘𝑃) − 1)))
44	43	bicomd 226	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → (1 < ((♯‘𝑃) − 1) ↔ (1 + 1) < (♯‘𝑃)))
45	21, 26, 19, 44	mp3an2i 1495	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) ↔ (1 + 1) < (♯‘𝑃)))
46		2re 12326	. . . . . . . . . . . . . . . . . 18 ⊢ 2 ∈ ℝ
47	38, 46	eqeltri 2861	. . . . . . . . . . . . . . . . 17 ⊢ (1 + 1) ∈ ℝ
48		ltle 11309	. . . . . . . . . . . . . . . . 17 ⊢ (((1 + 1) ∈ ℝ ∧ (♯‘𝑃) ∈ ℝ) → ((1 + 1) < (♯‘𝑃) → (1 + 1) ≤ (♯‘𝑃)))
49	47, 19, 48	sylancr 599	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝑃) ∈ ℕ₀ → ((1 + 1) < (♯‘𝑃) → (1 + 1) ≤ (♯‘𝑃)))
50	45, 49	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑃) ∈ ℕ₀ → (1 < ((♯‘𝑃) − 1) → (1 + 1) ≤ (♯‘𝑃)))
51	50	imp 412	. . . . . . . . . . . . . 14 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (1 + 1) ≤ (♯‘𝑃))
52		eluz2 12879	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)) ↔ ((1 + 1) ∈ ℤ ∧ (♯‘𝑃) ∈ ℤ ∧ (1 + 1) ≤ (♯‘𝑃)))
53	41, 42, 51, 52	syl3anbrc 1362	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑃) ∈ ℕ₀ ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)))
54	18, 53	sylan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1)))
55		fzosplitsnm1 13781	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ (♯‘𝑃) ∈ (ℤ_≥‘(1 + 1))) → (1..^(♯‘𝑃)) = ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))
56	37, 54, 55	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (1..^(♯‘𝑃)) = ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)}))
57	56	uneq2d 4122	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ({0} ∪ (1..^(♯‘𝑃))) = ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})))
58	36, 57	eqtrd 2800	. . . . . . . . 9 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (0..^(♯‘𝑃)) = ({0} ∪ ((1..^((♯‘𝑃) − 1)) ∪ {((♯‘𝑃) − 1)})))
59	58	raleqdv 3325	. . . . . . . 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 4150	. . . . . . . . 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 4150	. . . . . . . . . 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 635	. . . . . . . . 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 278	. . . . . . . 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 290	. . . . . . 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 2774	. . . . . . . . . . . 12 ⊢ ((♯‘𝑃) − 1) = 𝑅
66	65	oveq2i 7427	. . . . . . . . . . 11 ⊢ (1..^((♯‘𝑃) − 1)) = (1..^𝑅)
67	66	raleqi 3323	. . . . . . . . . 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 6904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (1..^𝑅) → ((𝑃 ↾ (1..^𝑅))‘𝑖) = (𝑃‘𝑖))
69	68	eqcomd 2771	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1..^𝑅) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
70	69	adantl 487	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
71	70	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → (𝑃‘𝑖) = ((𝑃 ↾ (1..^𝑅))‘𝑖))
72		fvres 6904	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (1..^𝑅) → ((𝑃 ↾ (1..^𝑅))‘𝑗) = (𝑃‘𝑗))
73	72	eqcomd 2771	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 ∈ (1..^𝑅) → (𝑃‘𝑗) = ((𝑃 ↾ (1..^𝑅))‘𝑗))
74	73	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) ∧ 𝑖 ∈ (1..^𝑅)) ∧ 𝑗 ∈ (1..^𝑅)) → (𝑃‘𝑗) = ((𝑃 ↾ (1..^𝑅))‘𝑗))
75	71, 74	neeq12d 3021	. . . . . . . . . . . . . 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 232	. . . . . . . . . . . . 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 58	. . . . . . . . . . . 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 3179	. . . . . . . . . . 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 3179	. . . . . . . . . 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 245	. . . . . . . . 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 497	. . . . . . . 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 496	. . . . . . 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 243	. . . . . 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 16	. . . . 5 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ∀𝑖 ∈ (1..^𝑅)∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → ((𝑃 ↾ (1..^𝑅))‘𝑖) ≠ ((𝑃 ↾ (1..^𝑅))‘𝑗)))
85		dff14a 7270	. . . . 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 595	. . . 4 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V)
87		df-f1 6545	. . . 4 ⊢ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)–1-1→V ↔ ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ Fun ^◡(𝑃 ↾ (1..^𝑅))))
88	86, 87	sylib 221	. . 3 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → ((𝑃 ↾ (1..^𝑅)):(1..^𝑅)⟶V ∧ Fun ^◡(𝑃 ↾ (1..^𝑅))))
89	88	simprd 501	. 2 ⊢ ((𝜑 ∧ 1 < ((♯‘𝑃) − 1)) → Fun ^◡(𝑃 ↾ (1..^𝑅)))
90		funcnv0 6606	. . 3 ⊢ Fun ^◡∅
91	18	nn0zd 12627	. . . . . . . . . . . . 13 ⊢ (𝜑 → (♯‘𝑃) ∈ ℤ)
92		peano2zm 12648	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑃) ∈ ℤ → ((♯‘𝑃) − 1) ∈ ℤ)
93	91, 92	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑃) − 1) ∈ ℤ)
94	93	zred 12711	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘𝑃) − 1) ∈ ℝ)
95		1red 11220	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
96	94, 95	lenltd 11367	. . . . . . . . . 10 ⊢ (𝜑 → (((♯‘𝑃) − 1) ≤ 1 ↔ ¬ 1 < ((♯‘𝑃) − 1)))
97	96	biimpar 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ((♯‘𝑃) − 1) ≤ 1)
98	5, 97	eqbrtrid 5148	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → 𝑅 ≤ 1)
99		1zzd 12636	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℤ)
100	5, 93	eqeltrid 2869	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℤ)
101	99, 100	jca 521	. . . . . . . . . 10 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑅 ∈ ℤ))
102	101	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (1 ∈ ℤ ∧ 𝑅 ∈ ℤ))
103		fzon 13721	. . . . . . . . . 10 ⊢ ((1 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑅 ≤ 1 ↔ (1..^𝑅) = ∅))
104	103	bicomd 226	. . . . . . . . 9 ⊢ ((1 ∈ ℤ ∧ 𝑅 ∈ ℤ) → ((1..^𝑅) = ∅ ↔ 𝑅 ≤ 1))
105	102, 104	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ((1..^𝑅) = ∅ ↔ 𝑅 ≤ 1))
106	98, 105	mpbird 260	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (1..^𝑅) = ∅)
107	106	reseq2d 5980	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)) = (𝑃 ↾ ∅))
108		res0 5984	. . . . . 6 ⊢ (𝑃 ↾ ∅) = ∅
109	107, 108	eqtrdi 2816	. . . . 5 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (𝑃 ↾ (1..^𝑅)) = ∅)
110	109	cnveqd 5863	. . . 4 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → ^◡(𝑃 ↾ (1..^𝑅)) = ^◡∅)
111	110	funeqd 6562	. . 3 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → (Fun ^◡(𝑃 ↾ (1..^𝑅)) ↔ Fun ^◡∅))
112	90, 111	mpbiri 261	. 2 ⊢ ((𝜑 ∧ ¬ 1 < ((♯‘𝑃) − 1)) → Fun ^◡(𝑃 ↾ (1..^𝑅)))
113	89, 112	pm2.61dan 825	1 ⊢ (𝜑 → Fun ^◡(𝑃 ↾ (1..^𝑅)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∀wral 3081 Vcvv 3457 ∪ cun 3904 ⊆ wss 3906 ∅c0 4286 {csn 4591 class class class wbr 5111 ^◡ccnv 5662 ↾ cres 5665 Fun wfun 6534 ⟶wf 6536 –1-1→wf1 6537 ‘cfv 6540 (class class class)co 7416 ℝcr 11110 0cc0 11111 1c1 11112 + caddc 11114 < clt 11254 ≤ cle 11255 − cmin 11452 ℕcn 12244 2c2 12306 ℕ₀cn0 12515 ℤcz 12602 ℤ_≥cuz 12873 ..^cfzo 13694 ♯chash 14379 Word cword 14563

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-card 9937 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-n0 12516 df-z 12603 df-uz 12874 df-fz 13547 df-fzo 13695 df-hash 14380 df-word 14564

This theorem is used by: pthd 30147

W3C validator