icceuelpart - Mathbox for Alexander van der Vekens

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Theorem icceuelpart 46104

Description: An element of a partitioned half-open 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 482	. . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → 𝑃 ∈ (RePart‘𝑀))
3		iccpartiun.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ)
4		iccelpart 46101	. . . . 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 482	. . 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 6891	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
8		fveq1 6891	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑀) = (𝑃‘𝑀))
9	7, 8	oveq12d 7427	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝‘𝑀)) = ((𝑃‘0)[,)(𝑃‘𝑀)))
10	9	eleq2d 2820	. . . . . . 7 ⊢ (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) ↔ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))))
11		fveq1 6891	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘𝑖) = (𝑃‘𝑖))
12		fveq1 6891	. . . . . . . . . 10 ⊢ (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
13	11, 12	oveq12d 7427	. . . . . . . . 9 ⊢ (𝑝 = 𝑃 → ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
14	13	eleq2d 2820	. . . . . . . 8 ⊢ (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1)))))
15	14	rexbidv 3179	. . . . . . 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 345	. . . . . 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 3611	. . . . 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 492	. . . 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 698	. 2 ⊢ ((𝜑 ∧ 𝑋 ∈ ((𝑃‘0)[,)(𝑃‘𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))))
21	3	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
22	1	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
23		elfzofz 13648	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
24	23	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
25	21, 22, 24	iccpartxr 46087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘𝑖) ∈ ℝ^*)
26		fzofzp1 13729	. . . . . . . . . . 11 ⊢ (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
27	26	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
28	21, 22, 27	iccpartxr 46087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ∈ ℝ^*)
29	25, 28	jca 513	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0..^𝑀)) → ((𝑃‘𝑖) ∈ ℝ^* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ^*))
30	29	adantrr 716	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘𝑖) ∈ ℝ^* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ^*))
31		elico1 13367	. . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
34	1	adantr 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
35		elfzofz 13648	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
36	35	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
37	33, 34, 36	iccpartxr 46087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑃‘𝑗) ∈ ℝ^*)
38		fzofzp1 13729	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
39	38	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
40	33, 34, 39	iccpartxr 46087	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → (𝑃‘(𝑗 + 1)) ∈ ℝ^*)
41	37, 40	jca 513	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑗 ∈ (0..^𝑀)) → ((𝑃‘𝑗) ∈ ℝ^* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ^*))
42	41	adantrl 715	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘𝑗) ∈ ℝ^* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ^*))
43		elico1 13367	. . . . . . 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 632	. . . . 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 13632	. . . . . . . . . 10 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
47	46	zred 12666	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
48		elfzoelz 13632	. . . . . . . . . 10 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
49	48	zred 12666	. . . . . . . . 9 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
50	47, 49	anim12i 614	. . . . . . . 8 ⊢ ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
51	50	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
52		lttri4 11298	. . . . . . 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 46103	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 < 𝑗 → (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗))))
55	54	imp31 419	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (𝑃‘(𝑖 + 1)) ≤ (𝑃‘𝑗))
56		simpl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → 𝑋 ∈ ℝ^*)
57	28	adantrr 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑖 + 1)) ∈ ℝ^*)
58	57	adantl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑖 + 1)) ∈ ℝ^*)
59	37	adantrl 715	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘𝑗) ∈ ℝ^*)
60	59	adantl 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘𝑗) ∈ ℝ^*)
61		nltle2tri 46021	. . . . . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 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 1136	. . . . . . . . . . 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 408	. . . . . . . . . 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 592	. . . . . . . 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 415	. . . . . . 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 46103	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖))))
79	78	ancomsd 467	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖))))
80	79	imp31 419	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (𝑃‘(𝑗 + 1)) ≤ (𝑃‘𝑖))
81	40	adantrl 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑗 + 1)) ∈ ℝ^*)
82	81	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑗 + 1)) ∈ ℝ^*)
83	25	adantrr 716	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘𝑖) ∈ ℝ^*)
84	83	adantl 483	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘𝑖) ∈ ℝ^*)
85		nltle2tri 46021	. . . . . . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . . . . . 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 121	. . . . . . . . . . . . . . . . . . 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 1354	. . . . . . . . . . . . . . . . . 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 414	. . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . 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 1135	. . . . . . . . . . 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 408	. . . . . . . . . 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 592	. . . . . . . 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 415	. . . . . . 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 1428	. . . . . 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 239	. . . 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 3201	. . 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 482	. 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 6892	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑃‘𝑖) = (𝑃‘𝑗))
106		fvoveq1 7432	. . . . 5 ⊢ (𝑖 = 𝑗 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝑗 + 1)))
107	105, 106	oveq12d 7427	. . . 4 ⊢ (𝑖 = 𝑗 → ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) = ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1))))
108	107	eleq2d 2820	. . 3 ⊢ (𝑖 = 𝑗 → (𝑋 ∈ ((𝑃‘𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃‘𝑗)[,)(𝑃‘(𝑗 + 1)))))
109	108	reu4 3728	. 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)))
110	20, 104, 109	sylanbrc 584	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 205 ∧ wa 397 ∨ w3o 1087 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∃!wreu 3375 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℕcn 12212 [,)cico 13326 ...cfz 13484 ..^cfzo 13627 RePartciccp 46081

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-ico 13330 df-fz 13485 df-fzo 13628 df-iccp 46082

This theorem is referenced by: iccpartdisj 46105

W3C validator