iccpartigtl - Mathbox for Alexander van der Vekens

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Theorem iccpartigtl 48708

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 4457	. . . 4 ⊢ ∀𝑖 ∈ ∅ (𝑃‘0) < (𝑃‘𝑖)
2		oveq2 7448	. . . . . 6 ⊢ (𝑀 = 1 → (1..^𝑀) = (1..^1))
3		fzo0 13842	. . . . . 6 ⊢ (1..^1) = ∅
4	2, 3	eqtrdi 2814	. . . . 5 ⊢ (𝑀 = 1 → (1..^𝑀) = ∅)
5	4	raleqdv 3323	. . . 4 ⊢ (𝑀 = 1 → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) ↔ ∀𝑖 ∈ ∅ (𝑃‘0) < (𝑃‘𝑖)))
6	1, 5	mpbiri 261	. . 3 ⊢ (𝑀 = 1 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
7	6	a1d 26	. 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 12691	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
11		0elfz 13782	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → 0 ∈ (0...𝑀))
12	10, 11	syl 18	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0...𝑀))
13	8, 9, 12	iccpartxr 48704	. . . . 5 ⊢ (𝜑 → (𝑃‘0) ∈ ℝ^*)
14	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑃‘0) ∈ ℝ^*)
15		elxr 13269	. . . . 5 ⊢ ((𝑃‘0) ∈ ℝ^* ↔ ((𝑃‘0) ∈ ℝ ∨ (𝑃‘0) = +∞ ∨ (𝑃‘0) = -∞))
16		0zd 12729	. . . . . . . . 9 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 0 ∈ ℤ)
17		elfzouz 13822	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (ℤ_≥‘1))
18		0p1e1 12487	. . . . . . . . . . . 12 ⊢ (0 + 1) = 1
19	18	fveq2i 6908	. . . . . . . . . . 11 ⊢ (ℤ_≥‘(0 + 1)) = (ℤ_≥‘1)
20	17, 19	eleqtrrdi 2875	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (ℤ_≥‘(0 + 1)))
21	20	adantl 487	. . . . . . . . 9 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (ℤ_≥‘(0 + 1)))
22		fveq2 6905	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 0 → (𝑃‘𝑘) = (𝑃‘0))
23	22	eqcomd 2769	. . . . . . . . . . . . 13 ⊢ (𝑘 = 0 → (𝑃‘0) = (𝑃‘𝑘))
24	23	eleq1d 2849	. . . . . . . . . . . 12 ⊢ (𝑘 = 0 → ((𝑃‘0) ∈ ℝ ↔ (𝑃‘𝑘) ∈ ℝ))
25	24	biimpcd 252	. . . . . . . . . . 11 ⊢ ((𝑃‘0) ∈ ℝ → (𝑘 = 0 → (𝑃‘𝑘) ∈ ℝ))
26	25	ad3antrrr 743	. . . . . . . . . 10 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 = 0 → (𝑃‘𝑘) ∈ ℝ))
27	8	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑀 ∈ ℕ)
28	9	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑃 ∈ (RePart‘𝑀))
29		elfz2nn0 13776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (0...𝑖) ↔ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖))
30		elfzo2 13820	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
31		simpl1 1210	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 ∈ ℕ₀)
32		simpr2 1214	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑀 ∈ ℤ)
33		nn0ge0 12655	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ ℕ₀ → 0 ≤ 𝑖)
34		0red 11335	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → 0 ∈ ℝ)
35		eluzelre 13000	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑖 ∈ (ℤ_≥‘1) → 𝑖 ∈ ℝ)
36	35	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → 𝑖 ∈ ℝ)
37		zre 12721	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
38	37	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℝ)
39		lelttr 11424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((0 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((0 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 0 < 𝑀))
40	34, 36, 38, 39	syl3anc 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → ((0 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 0 < 𝑀))
41	40	expcomd 422	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → (𝑖 < 𝑀 → (0 ≤ 𝑖 → 0 < 𝑀)))
42	41	3impia 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (0 ≤ 𝑖 → 0 < 𝑀))
43	33, 42	syl5com 32	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ ℕ₀ → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 0 < 𝑀))
44	43	3ad2ant2 1152	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 0 < 𝑀))
45	44	imp 412	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 0 < 𝑀)
46		elnnz 12727	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℕ ↔ (𝑀 ∈ ℤ ∧ 0 < 𝑀))
47	32, 45, 46	sylanbrc 595	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑀 ∈ ℕ)
48		nn0re 12639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑘 ∈ ℕ₀ → 𝑘 ∈ ℝ)
49	48	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → 𝑘 ∈ ℝ)
50		nn0re 12639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑖 ∈ ℕ₀ → 𝑖 ∈ ℝ)
51	50	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℝ)
52	51	adantl 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → 𝑖 ∈ ℝ)
53	38	adantr 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → 𝑀 ∈ ℝ)
54		lelttr 11424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → ((𝑘 ≤ 𝑖 ∧ 𝑖 < 𝑀) → 𝑘 < 𝑀))
55	54	expd 421	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀)))
56	49, 52, 53, 55	syl3anc 1399	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) ∧ (𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀)) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀)))
57	56	exp31 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑀 ∈ ℤ → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑘 ≤ 𝑖 → (𝑖 < 𝑀 → 𝑘 < 𝑀)))))
58	57	com34 92	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑀 ∈ ℤ → (𝑘 ≤ 𝑖 → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑖 < 𝑀 → 𝑘 < 𝑀)))))
59	58	com35 99	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑀 ∈ ℤ → (𝑖 < 𝑀 → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀)))))
60	59	3imp 1128	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀)))
61	60	expdcom 420	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (𝑘 ≤ 𝑖 → 𝑘 < 𝑀))))
62	61	com34 92	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ ℕ₀ → (𝑖 ∈ ℕ₀ → (𝑘 ≤ 𝑖 → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑘 < 𝑀))))
63	62	3imp1 1366	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 < 𝑀)
64		elfzo0 13859	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (0..^𝑀) ↔ (𝑘 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ ∧ 𝑘 < 𝑀))
65	31, 47, 63, 64	syl3anbrc 1362	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) ∧ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀)) → 𝑘 ∈ (0..^𝑀))
66	65	ex 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) → ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑘 ∈ (0..^𝑀)))
67	30, 66	biimtrid 245	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑘 ∈ ℕ₀ ∧ 𝑖 ∈ ℕ₀ ∧ 𝑘 ≤ 𝑖) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀)))
68	29, 67	sylbi 220	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (0...𝑖) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀)))
69	68	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑖 ∈ (1..^𝑀) → 𝑘 ∈ (0..^𝑀)))
70	69	impcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ∈ (0..^𝑀))
71		simpr 490	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → 𝑘 ≠ 0)
72	71	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ≠ 0)
73		fzo1fzo0n0 13874	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (0..^𝑀) ∧ 𝑘 ≠ 0))
74	70, 72, 73	sylanbrc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0)) → 𝑘 ∈ (1..^𝑀))
75	74	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → 𝑘 ∈ (1..^𝑀))
76	27, 28, 75	iccpartipre 48706	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑖 ∈ (1..^𝑀) ∧ (𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0))) → (𝑃‘𝑘) ∈ ℝ)
77	76	exp32 426	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑖 ∈ (1..^𝑀) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ)))
78	77	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (𝑖 ∈ (1..^𝑀) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ)))
79	78	imp 412	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ((𝑘 ∈ (0...𝑖) ∧ 𝑘 ≠ 0) → (𝑃‘𝑘) ∈ ℝ))
80	79	expdimp 458	. . . . . . . . . 10 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑘 ≠ 0 → (𝑃‘𝑘) ∈ ℝ))
81	26, 80	pm2.61dne 3045	. . . . . . . . 9 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...𝑖)) → (𝑃‘𝑘) ∈ ℝ)
82	8	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ∈ ℕ)
83	82	ad3antlr 744	. . . . . . . . . . 11 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑀 ∈ ℕ)
84	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
85	84	ad3antlr 744	. . . . . . . . . . 11 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑃 ∈ (RePart‘𝑀))
86		elfzoelz 13817	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ ℤ)
87	86	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ ℤ)
88		fzoval 13818	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ ℤ → (0..^𝑖) = (0...(𝑖 − 1)))
89	88	eqcomd 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ ℤ → (0...(𝑖 − 1)) = (0..^𝑖))
90	87, 89	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0...(𝑖 − 1)) = (0..^𝑖))
91	90	eleq2d 2850	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0...(𝑖 − 1)) ↔ 𝑘 ∈ (0..^𝑖)))
92		elfzouz2 13833	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ_≥‘𝑖))
93	92	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ (ℤ_≥‘𝑖))
94		fzoss2 13846	. . . . . . . . . . . . . . 15 ⊢ (𝑀 ∈ (ℤ_≥‘𝑖) → (0..^𝑖) ⊆ (0..^𝑀))
95	93, 94	syl 18	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0..^𝑖) ⊆ (0..^𝑀))
96	95	sseld 3912	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0..^𝑖) → 𝑘 ∈ (0..^𝑀)))
97	91, 96	sylbid 243	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑘 ∈ (0...(𝑖 − 1)) → 𝑘 ∈ (0..^𝑀)))
98	97	imp 412	. . . . . . . . . . 11 ⊢ (((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) ∧ 𝑘 ∈ (0...(𝑖 − 1))) → 𝑘 ∈ (0..^𝑀))
99		iccpartimp 48702	. . . . . . . . . . 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 1399	. . . . . . . . . 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 501	. . . . . . . . 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 48650	. . . . . . . 8 ⊢ ((((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘𝑖))
103	102	ralrimiva 3158	. . . . . . 7 ⊢ (((𝑃‘0) ∈ ℝ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
104	103	ex 418	. . . . . 6 ⊢ ((𝑃‘0) ∈ ℝ → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
105		lbfzo0 13858	. . . . . . . . . . . . . . . 16 ⊢ (0 ∈ (0..^𝑀) ↔ 𝑀 ∈ ℕ)
106	8, 105	sylibr 237	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ (0..^𝑀))
107	8, 9, 106	3jca 1146	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)))
108	107	ad2antrl 741	. . . . . . . . . . . . 13 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)))
109	108	adantr 486	. . . . . . . . . . . 12 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)))
110		iccpartimp 48702	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 0 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
111	109, 110	syl 18	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘0) < (𝑃‘(0 + 1))))
112	111	simprd 501	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘(0 + 1)))
113		breq1 5109	. . . . . . . . . . . 12 ⊢ ((𝑃‘0) = +∞ → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 + 1))))
114	113	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 + 1))))
115	114	adantr 486	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ((𝑃‘0) < (𝑃‘(0 + 1)) ↔ +∞ < (𝑃‘(0 + 1))))
116	112, 115	mpbid 235	. . . . . . . . 9 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → +∞ < (𝑃‘(0 + 1)))
117	8	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → 𝑀 ∈ ℕ)
118	117	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ)
119	9	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → 𝑃 ∈ (RePart‘𝑀))
120	119	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
121		1nn0 12646	. . . . . . . . . . . . . . . . . 18 ⊢ 1 ∈ ℕ₀
122	121	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 1 ∈ ℕ₀)
123		nnnn0 12637	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
124		nnge1 12390	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ ℕ → 1 ≤ 𝑀)
125	122, 123, 124	3jca 1146	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ ℕ → (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
126	8, 125	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
127		elfz2nn0 13776	. . . . . . . . . . . . . . 15 ⊢ (1 ∈ (0...𝑀) ↔ (1 ∈ ℕ₀ ∧ 𝑀 ∈ ℕ₀ ∧ 1 ≤ 𝑀))
128	126, 127	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 1 ∈ (0...𝑀))
129	18, 128	eqeltrid 2868	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0 + 1) ∈ (0...𝑀))
130	129	ad2antrl 741	. . . . . . . . . . . 12 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (0 + 1) ∈ (0...𝑀))
131	130	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (0 + 1) ∈ (0...𝑀))
132	118, 120, 131	iccpartxr 48704	. . . . . . . . . 10 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘(0 + 1)) ∈ ℝ^*)
133		pnfnlt 13281	. . . . . . . . . 10 ⊢ ((𝑃‘(0 + 1)) ∈ ℝ^* → ¬ +∞ < (𝑃‘(0 + 1)))
134	132, 133	syl 18	. . . . . . . . 9 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → ¬ +∞ < (𝑃‘(0 + 1)))
135	116, 134	pm2.21dd 198	. . . . . . . 8 ⊢ ((((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘0) < (𝑃‘𝑖))
136	135	ralrimiva 3158	. . . . . . 7 ⊢ (((𝑃‘0) = +∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
137	136	ex 418	. . . . . 6 ⊢ ((𝑃‘0) = +∞ → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
138	8	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ)
139	9	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
140		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (1..^𝑀))
141	138, 139, 140	iccpartipre 48706	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) ∈ ℝ)
142		mnflt 13276	. . . . . . . . . . 11 ⊢ ((𝑃‘𝑖) ∈ ℝ → -∞ < (𝑃‘𝑖))
143	141, 142	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1..^𝑀)) → -∞ < (𝑃‘𝑖))
144	143	ralrimiva 3158	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖))
145	144	ad2antrl 741	. . . . . . . 8 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖))
146		breq1 5109	. . . . . . . . . 10 ⊢ ((𝑃‘0) = -∞ → ((𝑃‘0) < (𝑃‘𝑖) ↔ -∞ < (𝑃‘𝑖)))
147	146	adantr 486	. . . . . . . . 9 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ((𝑃‘0) < (𝑃‘𝑖) ↔ -∞ < (𝑃‘𝑖)))
148	147	ralbidv 3189	. . . . . . . 8 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → (∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖) ↔ ∀𝑖 ∈ (1..^𝑀)-∞ < (𝑃‘𝑖)))
149	145, 148	mpbird 260	. . . . . . 7 ⊢ (((𝑃‘0) = -∞ ∧ (𝜑 ∧ ¬ 𝑀 = 1)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
150	149	ex 418	. . . . . 6 ⊢ ((𝑃‘0) = -∞ → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
151	104, 137, 150	3jaoi 1455	. . . . 5 ⊢ (((𝑃‘0) ∈ ℝ ∨ (𝑃‘0) = +∞ ∨ (𝑃‘0) = -∞) → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
152	15, 151	sylbi 220	. . . 4 ⊢ ((𝑃‘0) ∈ ℝ^* → ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
153	14, 152	mpcom 39	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))
154	153	expcom 419	. 2 ⊢ (¬ 𝑀 = 1 → (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖)))
155	7, 154	pm2.61i 184	1 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑖))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ⊆ wss 3900 ∅c0 4281 class class class wbr 5106 ‘cfv 6555 (class class class)co 7440 ↑_m cmap 8872 ℝcr 11223 0cc0 11224 1c1 11225 + caddc 11227 +∞cpnf 11364 -∞cmnf 11365 ℝ^*cxr 11366 < clt 11367 ≤ cle 11368 − cmin 11565 ℕcn 12359 ℕ₀cn0 12630 ℤcz 12717 ℤ_≥cuz 12989 ...cfz 13663 ..^cfzo 13812 RePartciccp 48698

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772 ax-cnex 11280 ax-resscn 11281 ax-1cn 11282 ax-icn 11283 ax-addcl 11284 ax-addrcl 11285 ax-mulcl 11286 ax-mulrcl 11287 ax-mulcom 11288 ax-addass 11289 ax-mulass 11290 ax-distr 11291 ax-i2m1 11292 ax-1ne0 11293 ax-1rid 11294 ax-rnegex 11295 ax-rrecex 11296 ax-cnre 11297 ax-pre-lttri 11298 ax-pre-lttrn 11299 ax-pre-ltadd 11300 ax-pre-mulgt0 11301

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6319 df-ord 6380 df-on 6381 df-lim 6382 df-suc 6383 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-riota 7397 df-ov 7443 df-oprab 7444 df-mpo 7445 df-om 7899 df-1st 8022 df-2nd 8023 df-frecs 8324 df-wrecs 8355 df-recs 8404 df-rdg 8443 df-er 8742 df-map 8874 df-en 8999 df-dom 9000 df-sdom 9001 df-pnf 11369 df-mnf 11370 df-xr 11371 df-ltxr 11372 df-le 11373 df-sub 11567 df-neg 11568 df-nn 12360 df-n0 12631 df-z 12718 df-uz 12990 df-fz 13664 df-fzo 13813 df-iccp 48699

This theorem is used by: iccpartlt 48709 iccpartgtl 48711 iccpartgt 48712

W3C validator