icceuelpart - Mathbox for Alexander van der Vekens

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Theorem icceuelpart 41372

Description: An element of a partitioned half opened interval of extended reals is an element of exactly one part of the partition. (Contributed by AV, 19-Jul-2020.)

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

**Assertion**
Ref	Expression
icceuelpart	⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))

Distinct variable groups: 𝑖,𝑀 𝑃,𝑖 𝑖,𝑋 𝜑,𝑖

Proof of Theorem icceuelpart Dummy variables 𝑗 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartiun.p	. . . 4 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
2	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → 𝑃 ∈ (RePart‘𝑀))
3		iccpartiun.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		iccelpart 41369	. . . . 5 ⊢ (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
5	3, 4	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
6	5	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
7		fveq1 6190	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
8		fveq1 6190	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑀) = (𝑃‘𝑀))
9	7, 8	oveq12d 6668	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝‘𝑀)) = ((𝑃‘0)[,)(𝑃‘𝑀)))
10	9	eleq2d 2687	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) ↔ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))))
11		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑖) = (𝑃‘𝑖))
12		fveq1 6190	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
13	11, 12	oveq12d 6668	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
14	13	eleq2d 2687	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
15	14	rexbidv 3052	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
16	10, 15	imbi12d 334	. . . . . 6 ⊢ (𝑝 = 𝑃 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))))
17	16	rspcva 3307	. . . . 5 ⊢ ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → (𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
18	17	adantld 483	. . . 4 ⊢ ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
19	18	com12 32	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
20	2, 6, 19	mp2and 715	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
21	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
22	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
23		elfzofz 12485	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
24	23	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
25	21, 22, 24	iccpartxr 41355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ∈ ℝ^*)
26		fzofzp1 12565	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
27	26	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
28	21, 22, 27	iccpartxr 41355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ∈ ℝ^*)
29	25, 28	jca 554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖) ∈ ℝ^* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ^*))
30	29	adantrr 753	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘𝑖) ∈ ℝ^* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ^*))
31		elico1 12218	. . . . . . 7 ⊢ (((𝑃‘𝑖) ∈ ℝ^* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ^) → (𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ (𝑋 ∈ ℝ^ ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1)))))
32	30, 31	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1)))))
33	3	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
34	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
35		elfzofz 12485	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
36	35	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
37	33, 34, 36	iccpartxr 41355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑃‘𝑗) ∈ ℝ^*)
38		fzofzp1 12565	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
39	38	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
40	33, 34, 39	iccpartxr 41355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑃‘(𝑗 + 1)) ∈ ℝ^*)
41	37, 40	jca 554	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑃‘𝑗) ∈ ℝ^* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ^*))
42	41	adantrl 752	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘𝑗) ∈ ℝ^* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ^*))
43		elico1 12218	. . . . . . 7 ⊢ (((𝑃‘𝑗) ∈ ℝ^* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ^) → (𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1))) ↔ (𝑋 ∈ ℝ^ ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))))
44	42, 43	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1))) ↔ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))))
45	32, 44	anbi12d 747	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))) ↔ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))))))
46		elfzoelz 12470	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
47	46	zred 11482	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
48		elfzoelz 12470	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
49	48	zred 11482	. . . . . . . . 9 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
50	47, 49	anim12i 590	. . . . . . . 8 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
51	50	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
52		lttri4 10122	. . . . . . 7 ⊢ ((𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑖 < 𝑗 ∨ 𝑖 = 𝑗 ∨ 𝑗 < 𝑖))
53	51, 52	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 < 𝑗 ∨ 𝑖 = 𝑗 ∨ 𝑗 < 𝑖))
54	3, 1	icceuelpartlem 41371	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 < 𝑗 → (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗))))
55	54	imp31 448	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗))
56		simpl 473	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → 𝑋 ∈ ℝ^*)
57	28	adantrr 753	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑖 + 1)) ∈ ℝ^*)
58	57	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑖 + 1)) ∈ ℝ^*)
59	37	adantrl 752	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘𝑗) ∈ ℝ^*)
60	59	adantl 482	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘𝑗) ∈ ℝ^*)
61		nltle2tri 41323	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ^* ∧ (𝑃‘𝑗) ∈ ℝ^*) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) ∧ (𝑃‘𝑗) ≤ 𝑋))
62	56, 58, 60, 61	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) ∧ (𝑃‘𝑗) ≤ 𝑋))
63	62	pm2.21d 118	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) ∧ (𝑃‘𝑗) ≤ 𝑋) → 𝑖 = 𝑗))
64	63	3expd 1284	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) → ((𝑃‘𝑗) ≤ 𝑋 → 𝑖 = 𝑗))))
65	64	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑋 ∈ ℝ^* → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) → ((𝑃‘𝑗) ≤ 𝑋 → 𝑖 = 𝑗)))))
66	65	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℝ^* → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) → ((𝑃‘𝑗) ≤ 𝑋 → 𝑖 = 𝑗)))))
67	66	com25 99	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ ℝ^* → ((𝑃‘𝑗) ≤ 𝑋 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗) → (𝑋 < (𝑃‘(𝑖 + 1)) → 𝑖 = 𝑗)))))
68	67	imp4b 613	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → (𝑋 < (𝑃‘(𝑖 + 1)) → 𝑖 = 𝑗)))
69	68	com23 86	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋) → (𝑋 < (𝑃‘(𝑖 + 1)) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → 𝑖 = 𝑗)))
70	69	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → (𝑋 < (𝑃‘(𝑖 + 1)) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → 𝑖 = 𝑗)))
71	70	com12 32	. . . . . . . . . . . 12 ⊢ (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → 𝑖 = 𝑗)))
72	71	3ad2ant3 1084	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) → ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → 𝑖 = 𝑗)))
73	72	imp 445	. . . . . . . . . 10 ⊢ (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → 𝑖 = 𝑗))
74	73	com12 32	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗)) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
75	55, 74	syldan 487	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
76	75	expcom 451	. . . . . . 7 ⊢ (𝑖 < 𝑗 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
77		2a1 28	. . . . . . 7 ⊢ (𝑖 = 𝑗 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
78	3, 1	icceuelpartlem 41371	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖))))
79	78	ancomsd 470	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖))))
80	79	imp31 448	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖))
81	40	adantrl 752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑗 + 1)) ∈ ℝ^*)
82	81	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑗 + 1)) ∈ ℝ^*)
83	25	adantrr 753	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘𝑖) ∈ ℝ^*)
84	83	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘𝑖) ∈ ℝ^*)
85		nltle2tri 41323	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ^* ∧ (𝑃‘𝑖) ∈ ℝ^*) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖) ∧ (𝑃‘𝑖) ≤ 𝑋))
86	56, 82, 84, 85	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖) ∧ (𝑃‘𝑖) ≤ 𝑋))
87	86	pm2.21d 118	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖) ∧ (𝑃‘𝑖) ≤ 𝑋) → 𝑖 = 𝑗))
88	87	3expd 1284	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖) → ((𝑃‘𝑖) ≤ 𝑋 → 𝑖 = 𝑗))))
89	88	ex 450	. . . . . . . . . . . . . . . . 17 ⊢ (𝑋 ∈ ℝ^* → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖) → ((𝑃‘𝑖) ≤ 𝑋 → 𝑖 = 𝑗)))))
90	89	com23 86	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ ℝ^* → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖) → ((𝑃‘𝑖) ≤ 𝑋 → 𝑖 = 𝑗)))))
91	90	imp4b 613	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → ((𝑃‘𝑖) ≤ 𝑋 → 𝑖 = 𝑗)))
92	91	com23 86	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ ℝ^* ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → ((𝑃‘𝑖) ≤ 𝑋 → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → 𝑖 = 𝑗)))
93	92	3adant2 1080	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → ((𝑃‘𝑖) ≤ 𝑋 → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → 𝑖 = 𝑗)))
94	93	com12 32	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑖) ≤ 𝑋 → ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → 𝑖 = 𝑗)))
95	94	3ad2ant2 1083	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) → ((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → 𝑖 = 𝑗)))
96	95	imp 445	. . . . . . . . . 10 ⊢ (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → 𝑖 = 𝑗))
97	96	com12 32	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖)) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
98	80, 97	syldan 487	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
99	98	expcom 451	. . . . . . 7 ⊢ (𝑗 < 𝑖 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
100	76, 77, 99	3jaoi 1391	. . . . . 6 ⊢ ((𝑖 < 𝑗 ∨ 𝑖 = 𝑗 ∨ 𝑗 < 𝑖) → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
101	53, 100	mpcom 38	. . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ^* ∧ (𝑃‘𝑖) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑃‘𝑗) ≤ 𝑋 ∧ 𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
102	45, 101	sylbid 230	. . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
103	102	ralrimivva 2971	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
104	103	adantr 481	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
105		fveq2 6191	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑃‘𝑖) = (𝑃‘𝑗))
106		oveq1 6657	. . . . . 6 ⊢ (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
107	106	fveq2d 6195	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝑗 + 1)))
108	105, 107	oveq12d 6668	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) = ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1))))
109	108	eleq2d 2687	. . 3 ⊢ (𝑖 = 𝑗 → (𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))))
110	109	reu4 3400	. 2 ⊢ (∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
111	20, 104, 110	sylanbrc 698	1 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 ∃!wreu 2914 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕcn 11020 [,)cico 12177 ...cfz 12326 ..^cfzo 12465 RePartciccp 41349

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-ico 12181 df-fz 12327 df-fzo 12466 df-iccp 41350

This theorem is referenced by: iccpartdisj 41373

W3C validator