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Theorem icceuelpart 42109
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 (𝜑𝑀 ∈ ℕ)
iccpartiun.p (𝜑𝑃 ∈ (RePart‘𝑀))
Assertion
Ref Expression
icceuelpart ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(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.
StepHypRef Expression
1 iccpartiun.p . . . 4 (𝜑𝑃 ∈ (RePart‘𝑀))
21adantr 472 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → 𝑃 ∈ (RePart‘𝑀))
3 iccpartiun.m . . . . 5 (𝜑𝑀 ∈ ℕ)
4 iccelpart 42106 . . . . 5 (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
53, 4syl 17 . . . 4 (𝜑 → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
65adantr 472 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
7 fveq1 6376 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
8 fveq1 6376 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑀) = (𝑃𝑀))
97, 8oveq12d 6862 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝𝑀)) = ((𝑃‘0)[,)(𝑃𝑀)))
109eleq2d 2830 . . . . . . 7 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) ↔ 𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))))
11 fveq1 6376 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
12 fveq1 6376 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
1311, 12oveq12d 6862 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
1413eleq2d 2830 . . . . . . . 8 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1514rexbidv 3199 . . . . . . 7 (𝑝 = 𝑃 → (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1610, 15imbi12d 335 . . . . . 6 (𝑝 = 𝑃 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))))
1716rspcva 3460 . . . . 5 ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → (𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1817adantld 484 . . . 4 ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1918com12 32 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ((𝑃 ∈ (RePart‘𝑀) ∧ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
202, 6, 19mp2and 690 . 2 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
213adantr 472 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
221adantr 472 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
23 elfzofz 12696 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
2423adantl 473 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
2521, 22, 24iccpartxr 42092 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃𝑖) ∈ ℝ*)
26 fzofzp1 12776 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
2726adantl 473 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2821, 22, 27iccpartxr 42092 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
2925, 28jca 507 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
3029adantrr 708 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
31 elico1 12423 . . . . . . 7 (((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*) → (𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1)))))
3230, 31syl 17 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1)))))
333adantr 472 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
341adantr 472 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
35 elfzofz 12696 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
3635adantl 473 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
3733, 34, 36iccpartxr 42092 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃𝑗) ∈ ℝ*)
38 fzofzp1 12776 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
3938adantl 473 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
4033, 34, 39iccpartxr 42092 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
4137, 40jca 507 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
4241adantrl 707 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
43 elico1 12423 . . . . . . 7 (((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*) → (𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))))
4442, 43syl 17 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1))) ↔ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))))
4532, 44anbi12d 624 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) ↔ ((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))))))
46 elfzoelz 12681 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
4746zred 11732 . . . . . . . . 9 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
48 elfzoelz 12681 . . . . . . . . . 10 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
4948zred 11732 . . . . . . . . 9 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
5047, 49anim12i 606 . . . . . . . 8 ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
5150adantl 473 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
52 lttri4 10378 . . . . . . 7 ((𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ) → (𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖))
5351, 52syl 17 . . . . . 6 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖))
543, 1icceuelpartlem 42108 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 < 𝑗 → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))))
5554imp31 408 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))
56 simpl 474 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → 𝑋 ∈ ℝ*)
5728adantrr 708 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
5857adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
5937adantrl 707 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃𝑗) ∈ ℝ*)
6059adantl 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃𝑗) ∈ ℝ*)
61 nltle2tri 42060 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ* ∧ (𝑃𝑗) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6256, 58, 60, 61syl3anc 1490 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6362pm2.21d 119 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋) → 𝑖 = 𝑗))
64633expd 1462 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → ((𝑃𝑗) ≤ 𝑋𝑖 = 𝑗))))
6564ex 401 . . . . . . . . . . . . . . . . . 18 (𝑋 ∈ ℝ* → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → ((𝑃𝑗) ≤ 𝑋𝑖 = 𝑗)))))
6665com23 86 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ℝ* → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → ((𝑃𝑗) ≤ 𝑋𝑖 = 𝑗)))))
6766com25 99 . . . . . . . . . . . . . . . 16 (𝑋 ∈ ℝ* → ((𝑃𝑗) ≤ 𝑋 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → (𝑋 < (𝑃‘(𝑖 + 1)) → 𝑖 = 𝑗)))))
6867imp4b 412 . . . . . . . . . . . . . . 15 ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → (𝑋 < (𝑃‘(𝑖 + 1)) → 𝑖 = 𝑗)))
6968com23 86 . . . . . . . . . . . . . 14 ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋) → (𝑋 < (𝑃‘(𝑖 + 1)) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → 𝑖 = 𝑗)))
70693adant3 1162 . . . . . . . . . . . . 13 ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))) → (𝑋 < (𝑃‘(𝑖 + 1)) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → 𝑖 = 𝑗)))
7170com12 32 . . . . . . . . . . . 12 (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → 𝑖 = 𝑗)))
72713ad2ant3 1165 . . . . . . . . . . 11 ((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) → ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → 𝑖 = 𝑗)))
7372imp 395 . . . . . . . . . 10 (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → 𝑖 = 𝑗))
7473com12 32 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗)) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
7555, 74syldan 585 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
7675expcom 402 . . . . . . 7 (𝑖 < 𝑗 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
77 2a1 28 . . . . . . 7 (𝑖 = 𝑗 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
783, 1icceuelpartlem 42108 . . . . . . . . . . 11 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
7978ancomsd 457 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
8079imp31 408 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))
8140adantrl 707 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
8281adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
8325adantrr 708 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃𝑖) ∈ ℝ*)
8483adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃𝑖) ∈ ℝ*)
85 nltle2tri 42060 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ* ∧ (𝑃𝑖) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8656, 82, 84, 85syl3anc 1490 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8786pm2.21d 119 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋) → 𝑖 = 𝑗))
88873expd 1462 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗))))
8988ex 401 . . . . . . . . . . . . . . . . 17 (𝑋 ∈ ℝ* → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗)))))
9089com23 86 . . . . . . . . . . . . . . . 16 (𝑋 ∈ ℝ* → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗)))))
9190imp4b 412 . . . . . . . . . . . . . . 15 ((𝑋 ∈ ℝ*𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗)))
9291com23 86 . . . . . . . . . . . . . 14 ((𝑋 ∈ ℝ*𝑋 < (𝑃‘(𝑗 + 1))) → ((𝑃𝑖) ≤ 𝑋 → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → 𝑖 = 𝑗)))
93923adant2 1161 . . . . . . . . . . . . 13 ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))) → ((𝑃𝑖) ≤ 𝑋 → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → 𝑖 = 𝑗)))
9493com12 32 . . . . . . . . . . . 12 ((𝑃𝑖) ≤ 𝑋 → ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → 𝑖 = 𝑗)))
95943ad2ant2 1164 . . . . . . . . . . 11 ((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) → ((𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → 𝑖 = 𝑗)))
9695imp 395 . . . . . . . . . 10 (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → 𝑖 = 𝑗))
9796com12 32 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖)) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
9880, 97syldan 585 . . . . . . . 8 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
9998expcom 402 . . . . . . 7 (𝑗 < 𝑖 → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
10076, 77, 993jaoi 1552 . . . . . 6 ((𝑖 < 𝑗𝑖 = 𝑗𝑗 < 𝑖) → ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
10153, 100mpcom 38 . . . . 5 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (((𝑋 ∈ ℝ* ∧ (𝑃𝑖) ≤ 𝑋𝑋 < (𝑃‘(𝑖 + 1))) ∧ (𝑋 ∈ ℝ* ∧ (𝑃𝑗) ≤ 𝑋𝑋 < (𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
10245, 101sylbid 231 . . . 4 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
103102ralrimivva 3118 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
104103adantr 472 . 2 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
105 fveq2 6377 . . . . 5 (𝑖 = 𝑗 → (𝑃𝑖) = (𝑃𝑗))
106 fvoveq1 6867 . . . . 5 (𝑖 = 𝑗 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝑗 + 1)))
107105, 106oveq12d 6862 . . . 4 (𝑖 = 𝑗 → ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) = ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1))))
108107eleq2d 2830 . . 3 (𝑖 = 𝑗 → (𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))))
109108reu4 3561 . 2 (∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗)))
11020, 104, 109sylanbrc 578 1 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃!𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3o 1106  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  ∃!wreu 3057   class class class wbr 4811  cfv 6070  (class class class)co 6844  cr 10190  0cc0 10191  1c1 10192   + caddc 10194  *cxr 10329   < clt 10330  cle 10331  cn 11276  [,)cico 12382  ...cfz 12536  ..^cfzo 12676  RePartciccp 42086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-nn 11277  df-2 11337  df-n0 11541  df-z 11627  df-uz 11890  df-ico 12386  df-fz 12537  df-fzo 12677  df-iccp 42087
This theorem is referenced by:  iccpartdisj  42110
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