iccpartigtl - Mathbox for Alexander van der Vekens

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Theorem iccpartigtl 44875

Description: If there is a partition, then all intermediate points are strictly greater than the lower bound. (Contributed by AV, 12-Jul-2020.)

**Hypotheses**
Ref	Expression
iccpartgtprec.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
iccpartgtprec.p	⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))

**Assertion**
Ref	Expression
iccpartigtl	⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))

Distinct variable groups: 𝑖,𝑀 𝑃,𝑖 𝜑,𝑖

Proof of Theorem iccpartigtl Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ral0 4443	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑃‘0) < (𝑃‘𝑖)
2		oveq2 7283	. . . . . 6 ⊢ (𝑀 = 1 → (1..^𝑀) = (1..^1))
3		fzo0 13411	. . . . . 6 ⊢ (1..^1) = ∅
4	2, 3	eqtrdi 2794	. . . . 5 ⊢ (𝑀 = 1 → (1..^𝑀) = ∅)
5	4	raleqdv 3348	. . . 4 ⊢ (𝑀 = 1 → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝑃‘0) < (𝑃‘𝑖)))
6	1, 5	mpbiri 257	. . 3 ⊢ (𝑀 = 1 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
7	6	a1d 25	. 2 ⊢ (𝑀 = 1 → (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
8		iccpartgtprec.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
9		iccpartgtprec.p	. . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
10	8	nnnn0d 12293	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
11		0elfz 13353	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
12	10, 11	syl 17	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
13	8, 9, 12	iccpartxr 44871	. . . . 5 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
14	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈ ℝ^*)
15		elxr 12852	. . . . 5 ⊢ ((𝑃‘0) ∈ ℝ^* ↔ ((𝑃‘0) ∈ ℝ ∨ (𝑃‘0) = +∞ ∨ (𝑃‘0) = -∞))
16		0zd 12331	. . . . . . . . 9 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 0 ∈ ℤ)
17		elfzouz 13391	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (ℤ_≥‘1))
18		0p1e1 12095	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
19	18	fveq2i 6777	. . . . . . . . . . 11 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
20	17, 19	eleqtrrdi 2850	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (ℤ_≥‘(0 + 1)))
21	20	adantl 482	. . . . . . . . 9 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (ℤ_≥‘(0 + 1)))
22		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
23	22	eqcomd 2744	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑃‘0) = (𝑃‘𝑘))
24	23	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑃‘0) ∈ ℝ ↔ (𝑃‘𝑘) ∈ ℝ))
25	24	biimpcd 248	. . . . . . . . . . 11 ⊢ ((𝑃‘0) ∈ ℝ → (𝑘 = 0 → (𝑃‘𝑘) ∈ ℝ))
26	25	ad3antrrr 727	. . . . . . . . . 10 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 = 0 → (𝑃‘𝑘) ∈ ℝ))
27	8	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑀 ∈ ℕ)
28	9	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑃 ∈ (RePart‘𝑀))
29		elfz2nn0 13347	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...𝑖) ↔ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖))
30		elfzo2 13390	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
31		simpl1 1190	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 ∈ ℕ₀)
32		simpr2 1194	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑀 ∈ ℤ)
33		nn0ge0 12258	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ ℕ₀ → 0 ≤ 𝑖)
34		0red 10978	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → 0 ∈ ℝ)
35		eluzelre 12593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ∈ (ℤ_≥‘1) → 𝑖 ∈ ℝ)
36	35	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → 𝑖 ∈ ℝ)
37		zre 12323	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
38	37	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ)
39		lelttr 11065	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 0 < 𝑀))
40	34, 36, 38, 39	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → ((0 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 0 < 𝑀))
41	40	expcomd 417	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → (𝑖 < 𝑀 → (0 ≤ 𝑖 → 0 < 𝑀)))
42	41	3impia 1116	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (0 ≤ 𝑖 → 0 < 𝑀))
43	33, 42	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ ℕ₀ → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 0 < 𝑀))
44	43	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 0 < 𝑀))
45	44	imp 407	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 0 < 𝑀)
46		elnnz 12329	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 0 < 𝑀))
47	32, 45, 46	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑀 ∈ ℕ)
48		nn0re 12242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
49	48	ad2antrl 725	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → 𝑘 ∈ ℝ)
50		nn0re 12242	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
51	50	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
52	51	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → 𝑖 ∈ ℝ)
53	38	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → 𝑀 ∈ ℝ)
54		lelttr 11065	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((𝑘 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 𝑘 < 𝑀))
55	54	expd 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀)))
56	49, 52, 53, 55	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀)))
57	56	exp31 420	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑀 ∈ ℤ → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀)))))
58	57	com34 91	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑀 ∈ ℤ → (𝑘 ≤ 𝑖 → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑖 < 𝑀 → 𝑘 < 𝑀)))))
59	58	com35 98	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑀 ∈ ℤ → (𝑖 < 𝑀 → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀)))))
60	59	3imp 1110	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀)))
61	60	expdcom 415	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀))))
62	61	com34 91	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ → (𝑘 ≤ 𝑖 → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑘 < 𝑀))))
63	62	3imp1 1346	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 < 𝑀)
64		elfzo0 13428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0..^𝑀) ↔ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ 𝑘 < 𝑀))
65	31, 47, 63, 64	syl3anbrc 1342	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 ∈ (0..^𝑀))
66	65	ex 413	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑘 ∈ (0..^𝑀)))
67	30, 66	syl5bi 241	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀)))
68	29, 67	sylbi 216	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...𝑖) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀)))
69	68	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀)))
70	69	impcom 408	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ∈ (0..^𝑀))
71		simpr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → 𝑘 ≠ 0)
72	71	adantl 482	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ≠ 0)
73		fzo1fzo0n0 13438	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (0..^𝑀) ∧ 𝑘 ≠ 0))
74	70, 72, 73	sylanbrc 583	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ∈ (1..^𝑀))
75	74	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑘 ∈ (1..^𝑀))
76	27, 28, 75	iccpartipre 44873	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → (𝑃‘𝑘) ∈ ℝ)
77	76	exp32 421	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑖 ∈ (1..^𝑀) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ)))
78	77	ad2antrl 725	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (𝑖 ∈ (1..^𝑀) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ)))
79	78	imp 407	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ))
80	79	expdimp 453	. . . . . . . . . 10 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 ≠ 0 → (𝑃‘𝑘) ∈ ℝ))
81	26, 80	pm2.61dne 3031	. . . . . . . . 9 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑃‘𝑘) ∈ ℝ)
82	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ)
83	82	ad3antlr 728	. . . . . . . . . . 11 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℕ)
84	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
85	84	ad3antlr 728	. . . . . . . . . . 11 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑃 ∈ (RePart‘𝑀))
86		elfzoelz 13387	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ ℤ)
87	86	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ ℤ)
88		fzoval 13388	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0..^𝑖) = (0...(𝑖 − 1)))
89	88	eqcomd 2744	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0...(𝑖 − 1)) = (0..^𝑖))
90	87, 89	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0...(𝑖 − 1)) = (0..^𝑖))
91	90	eleq2d 2824	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0...(𝑖 − 1)) ↔ 𝑘 ∈ (0..^𝑖)))
92		elfzouz2 13402	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ_≥‘𝑖))
93	92	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑖))
94		fzoss2 13415	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘𝑖) → (0..^𝑖) ⊆ (0..^𝑀))
95	93, 94	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0..^𝑖) ⊆ (0..^𝑀))
96	95	sseld 3920	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0..^𝑖) → 𝑘 ∈ (0..^𝑀)))
97	91, 96	sylbid 239	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0...(𝑖 − 1)) → 𝑘 ∈ (0..^𝑀)))
98	97	imp 407	. . . . . . . . . . 11 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑘 ∈ (0..^𝑀))
99		iccpartimp 44869	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))))
100	83, 85, 98, 99	syl3anc 1370	. . . . . . . . . 10 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))))
101	100	simprd 496	. . . . . . . . 9 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → (𝑃‘𝑘) < (𝑃‘(𝑘 + 1)))
102	16, 21, 81, 101	smonoord 44823	. . . . . . . 8 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘𝑖))
103	102	ralrimiva 3103	. . . . . . 7 ⊢ (((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
104	103	ex 413	. . . . . 6 ⊢ ((𝑃‘0) ∈ ℝ → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
105		lbfzo0 13427	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
106	8, 105	sylibr 233	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
107	8, 9, 106	3jca 1127	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)))
108	107	ad2antrl 725	. . . . . . . . . . . . 13 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)))
109	108	adantr 481	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)))
110		iccpartimp 44869	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
111	109, 110	syl 17	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
112	111	simprd 496	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘(0 + 1)))
113		breq1 5077	. . . . . . . . . . . 12 ⊢ ((𝑃‘0) = +∞ → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 + 1))))
114	113	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 + 1))))
115	114	adantr 481	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 + 1))))
116	112, 115	mpbid 231	. . . . . . . . 9 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → +∞ < (𝑃‘(0 + 1)))
117	8	ad2antrl 725	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → 𝑀 ∈ ℕ)
118	117	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ)
119	9	ad2antrl 725	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → 𝑃 ∈ (RePart‘𝑀))
120	119	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
121		1nn0 12249	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
122	121	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℕ₀)
123		nnnn0 12240	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
124		nnge1 12001	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
125	122, 123, 124	3jca 1127	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
126	8, 125	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
127		elfz2nn0 13347	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
128	126, 127	sylibr 233	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ (0...𝑀))
129	18, 128	eqeltrid 2843	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 + 1) ∈ (0...𝑀))
130	129	ad2antrl 725	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (0 + 1) ∈ (0...𝑀))
131	130	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0 + 1) ∈ (0...𝑀))
132	118, 120, 131	iccpartxr 44871	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘(0 + 1)) ∈ ℝ^*)
133		pnfnlt 12864	. . . . . . . . . 10 ⊢ ((𝑃‘(0 + 1)) ∈ ℝ^* → ¬ +∞ < (𝑃‘(0 + 1)))
134	132, 133	syl 17	. . . . . . . . 9 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ¬ +∞ < (𝑃‘(0 + 1)))
135	116, 134	pm2.21dd 194	. . . . . . . 8 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘𝑖))
136	135	ralrimiva 3103	. . . . . . 7 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
137	136	ex 413	. . . . . 6 ⊢ ((𝑃‘0) = +∞ → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
138	8	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ)
139	9	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
140		simpr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (1..^𝑀))
141	138, 139, 140	iccpartipre 44873	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) ∈ ℝ)
142		mnflt 12859	. . . . . . . . . . 11 ⊢ ((𝑃‘𝑖) ∈ ℝ → -∞ < (𝑃‘𝑖))
143	141, 142	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → -∞ < (𝑃‘𝑖))
144	143	ralrimiva 3103	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖))
145	144	ad2antrl 725	. . . . . . . 8 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖))
146		breq1 5077	. . . . . . . . . 10 ⊢ ((𝑃‘0) = -∞ → ((𝑃‘0) < (𝑃‘𝑖) ↔ -∞ < (𝑃‘𝑖)))
147	146	adantr 481	. . . . . . . . 9 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ((𝑃‘0) < (𝑃‘𝑖) ↔ -∞ < (𝑃‘𝑖)))
148	147	ralbidv 3112	. . . . . . . 8 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) ↔ ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖)))
149	145, 148	mpbird 256	. . . . . . 7 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
150	149	ex 413	. . . . . 6 ⊢ ((𝑃‘0) = -∞ → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
151	104, 137, 150	3jaoi 1426	. . . . 5 ⊢ (((𝑃‘0) ∈ ℝ ∨ (𝑃‘0) = +∞ ∨ (𝑃‘0) = -∞) → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
152	15, 151	sylbi 216	. . . 4 ⊢ ((𝑃‘0) ∈ ℝ^* → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
153	14, 152	mpcom 38	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
154	153	expcom 414	. 2 ⊢ (¬ 𝑀 = 1 → (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
155	7, 154	pm2.61i 182	1 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ w3o 1085 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⊆ wss 3887 ∅c0 4256 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 +∞cpnf 11006 -∞cmnf 11007 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 ℕcn 11973 ℕ₀cn0 12233 ℤcz 12319 ℤ_≥cuz 12582 ...cfz 13239 ..^cfzo 13382 RePartciccp 44865

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-iccp 44866

This theorem is referenced by: iccpartlt 44876 iccpartgtl 44878 iccpartgt 44879

W3C validator