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Theorem icceuelpart 41697
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 480 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → 𝑃 ∈ (RePart‘𝑀))
3 iccpartiun.m . . . . 5 (𝜑𝑀 ∈ ℕ)
4 iccelpart 41694 . . . . 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 480 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1)))))
7 fveq1 6228 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
8 fveq1 6228 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑀) = (𝑃𝑀))
97, 8oveq12d 6708 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝𝑀)) = ((𝑃‘0)[,)(𝑃𝑀)))
109eleq2d 2716 . . . . . . 7 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) ↔ 𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))))
11 fveq1 6228 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
12 fveq1 6228 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
1311, 12oveq12d 6708 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
1413eleq2d 2716 . . . . . . . 8 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1514rexbidv 3081 . . . . . . 7 (𝑝 = 𝑃 → (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1610, 15imbi12d 333 . . . . . 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 3338 . . . . 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 482 . . . 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 715 . 2 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
213adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
221adantr 480 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
23 elfzofz 12524 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
2423adantl 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
2521, 22, 24iccpartxr 41680 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃𝑖) ∈ ℝ*)
26 fzofzp1 12605 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
2726adantl 481 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2821, 22, 27iccpartxr 41680 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
2925, 28jca 553 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
3029adantrr 753 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
31 elico1 12256 . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑀 ∈ ℕ)
341adantr 480 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
35 elfzofz 12524 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
3635adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
3733, 34, 36iccpartxr 41680 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃𝑗) ∈ ℝ*)
38 fzofzp1 12605 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
3938adantl 481 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
4033, 34, 39iccpartxr 41680 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
4137, 40jca 553 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
4241adantrl 752 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
43 elico1 12256 . . . . . . 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 747 . . . . 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 12509 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
4746zred 11520 . . . . . . . . 9 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
48 elfzoelz 12509 . . . . . . . . . 10 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
4948zred 11520 . . . . . . . . 9 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
5047, 49anim12i 589 . . . . . . . 8 ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
5150adantl 481 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑖 ∈ ℝ ∧ 𝑗 ∈ ℝ))
52 lttri4 10160 . . . . . . 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 41696 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 < 𝑗 → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))))
5554imp31 447 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))
56 simpl 472 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → 𝑋 ∈ ℝ*)
5728adantrr 753 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
5857adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
5937adantrl 752 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃𝑗) ∈ ℝ*)
6059adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃𝑗) ∈ ℝ*)
61 nltle2tri 41648 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ* ∧ (𝑃𝑗) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6256, 58, 60, 61syl3anc 1366 . . . . . . . . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → ((𝑃𝑗) ≤ 𝑋𝑖 = 𝑗))))
6564ex 449 . . . . . . . . . . . . . . . . . 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 1101 . . . . . . . . . . . . 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 1104 . . . . . . . . . . 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 444 . . . . . . . . . 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 486 . . . . . . . 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 450 . . . . . . 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 41696 . . . . . . . . . . 11 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
7978ancomsd 469 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
8079imp31 447 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))
8140adantrl 752 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
8281adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
8325adantrr 753 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → (𝑃𝑖) ∈ ℝ*)
8483adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑃𝑖) ∈ ℝ*)
85 nltle2tri 41648 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ* ∧ (𝑃𝑖) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8656, 82, 84, 85syl3anc 1366 . . . . . . . . . . . . . . . . . . . 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 1306 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗))))
8988ex 449 . . . . . . . . . . . . . . . . 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 1100 . . . . . . . . . . . . 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 1103 . . . . . . . . . . 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 444 . . . . . . . . . 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 486 . . . . . . . 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 450 . . . . . . 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 1431 . . . . . 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 3000 . . 3 (𝜑 → ∀𝑖 ∈ (0..^𝑀)∀𝑗 ∈ (0..^𝑀)((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
104103adantr 480 . 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 6229 . . . . 5 (𝑖 = 𝑗 → (𝑃𝑖) = (𝑃𝑗))
106 oveq1 6697 . . . . . 6 (𝑖 = 𝑗 → (𝑖 + 1) = (𝑗 + 1))
107106fveq2d 6233 . . . . 5 (𝑖 = 𝑗 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝑗 + 1)))
108105, 107oveq12d 6708 . . . 4 (𝑖 = 𝑗 → ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) = ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1))))
109108eleq2d 2716 . . 3 (𝑖 = 𝑗 → (𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))))
110109reu4 3433 . 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 699 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 383  w3o 1053  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943   class class class wbr 4685  cfv 5926  (class class class)co 6690  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cn 11058  [,)cico 12215  ...cfz 12364  ..^cfzo 12504  RePartciccp 41674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-n0 11331  df-z 11416  df-uz 11726  df-ico 12219  df-fz 12365  df-fzo 12505  df-iccp 41675
This theorem is referenced by:  iccpartdisj  41698
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