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Theorem icceuelpart 40696
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 (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 481 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → 𝑃 ∈ (RePart‘𝑀))
3 iccpartiun.m . . . . 5 (𝜑𝑀 ∈ ℕ)
4 iccelpart 40693 . . . . 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 481 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
7 fveq1 6152 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
8 fveq1 6152 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑀) = (𝑃𝑀))
97, 8oveq12d 6628 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝𝑀)) = ((𝑃‘0)[,)(𝑃𝑀)))
109eleq2d 2684 . . . . . . 7 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) ↔ 𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))))
11 fveq1 6152 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
12 fveq1 6152 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
1311, 12oveq12d 6628 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
1413eleq2d 2684 . . . . . . . 8 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1514rexbidv 3046 . . . . . . 7 (𝑝 = 𝑃 → (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1610, 15imbi12d 334 . . . . . 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 3296 . . . . 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 483 . . . 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 714 . 2 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
213adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
221adantr 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
23 elfzofz 12434 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
2423adantl 482 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
2521, 22, 24iccpartxr 40679 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃𝑖) ∈ ℝ*)
26 fzofzp1 12514 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
2726adantl 482 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2821, 22, 27iccpartxr 40679 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
2925, 28jca 554 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
3029adantrr 752 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
31 elico1 12168 . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
341adantr 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
35 elfzofz 12434 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
3635adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
3733, 34, 36iccpartxr 40679 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃𝑗) ∈ ℝ*)
38 fzofzp1 12514 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
3938adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
4033, 34, 39iccpartxr 40679 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
4137, 40jca 554 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
4241adantrl 751 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
43 elico1 12168 . . . . . . 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 746 . . . . 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 12419 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
4746zred 11434 . . . . . . . . 9 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
48 elfzoelz 12419 . . . . . . . . . 10 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
4948zred 11434 . . . . . . . . 9 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
5047, 49anim12i 589 . . . . . . . 8 ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
5150adantl 482 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
52 lttri4 10074 . . . . . . 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 40695 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 < 𝑗 → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))))
5554imp31 448 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))
56 simpl 473 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → 𝑋 ∈ ℝ*)
5728adantrr 752 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
5857adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
5937adantrl 751 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃𝑗) ∈ ℝ*)
6059adantl 482 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃𝑗) ∈ ℝ*)
61 nltle2tri 40646 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ* ∧ (𝑃𝑗) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6256, 58, 60, 61syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6362pm2.21d 118 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋) → 𝑖 = 𝑗))
64633expd 1281 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → ((𝑃𝑗) ≤ 𝑋𝑖 = 𝑗))))
6564ex 450 . . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . 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 1079 . . . . . . . . . . . . 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 1082 . . . . . . . . . . 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 445 . . . . . . . . . 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 487 . . . . . . . 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 451 . . . . . . 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 40695 . . . . . . . . . . 11 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
7978ancomsd 470 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
8079imp31 448 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))
8140adantrl 751 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
8281adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
8325adantrr 752 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃𝑖) ∈ ℝ*)
8483adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃𝑖) ∈ ℝ*)
85 nltle2tri 40646 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ* ∧ (𝑃𝑖) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8656, 82, 84, 85syl3anc 1323 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8786pm2.21d 118 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋) → 𝑖 = 𝑗))
88873expd 1281 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗))))
8988ex 450 . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . 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 1078 . . . . . . . . . . . . 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 1081 . . . . . . . . . . 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 445 . . . . . . . . . 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 487 . . . . . . . 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 451 . . . . . . 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 1388 . . . . . 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 230 . . . 4 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
103102ralrimivva 2966 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
104103adantr 481 . 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 6153 . . . . 5 (𝑖 = 𝑗 → (𝑃𝑖) = (𝑃𝑗))
106 oveq1 6617 . . . . . 6 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
107106fveq2d 6157 . . . . 5 (𝑖 = 𝑗 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝑗 + 1)))
108105, 107oveq12d 6628 . . . 4 (𝑖 = 𝑗 → ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) = ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1))))
109108eleq2d 2684 . . 3 (𝑖 = 𝑗 → (𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))))
110109reu4 3386 . 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)))
11120, 104, 110sylanbrc 697 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 196  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909   class class class wbr 4618  cfv 5852  (class class class)co 6610  cr 9887  0cc0 9888  1c1 9889   + caddc 9891  *cxr 10025   < clt 10026  cle 10027  cn 10972  [,)cico 12127  ...cfz 12276  ..^cfzo 12414  RePartciccp 40673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-nn 10973  df-2 11031  df-n0 11245  df-z 11330  df-uz 11640  df-ico 12131  df-fz 12277  df-fzo 12415  df-iccp 40674
This theorem is referenced by:  iccpartdisj  40697
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