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Theorem icceuelpart 45618
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 481 . . 3 ((𝜑𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))) → 𝑃 ∈ (RePart‘𝑀))
3 iccpartiun.m . . . . 5 (𝜑𝑀 ∈ ℕ)
4 iccelpart 45615 . . . . 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 6841 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘0) = (𝑃‘0))
8 fveq1 6841 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑀) = (𝑃𝑀))
97, 8oveq12d 7375 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝‘0)[,)(𝑝𝑀)) = ((𝑃‘0)[,)(𝑃𝑀)))
109eleq2d 2823 . . . . . . 7 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝𝑀)) ↔ 𝑋 ∈ ((𝑃‘0)[,)(𝑃𝑀))))
11 fveq1 6841 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝𝑖) = (𝑃𝑖))
12 fveq1 6841 . . . . . . . . . 10 (𝑝 = 𝑃 → (𝑝‘(𝑖 + 1)) = (𝑃‘(𝑖 + 1)))
1311, 12oveq12d 7375 . . . . . . . . 9 (𝑝 = 𝑃 → ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))))
1413eleq2d 2823 . . . . . . . 8 (𝑝 = 𝑃 → (𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1514rexbidv 3175 . . . . . . 7 (𝑝 = 𝑃 → (∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1)))))
1610, 15imbi12d 344 . . . . . 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 3579 . . . . 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 491 . . . 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 697 . 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 13588 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ (0...𝑀))
2423adantl 482 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → 𝑖 ∈ (0...𝑀))
2521, 22, 24iccpartxr 45601 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃𝑖) ∈ ℝ*)
26 fzofzp1 13669 . . . . . . . . . . 11 (𝑖 ∈ (0..^𝑀) → (𝑖 + 1) ∈ (0...𝑀))
2726adantl 482 . . . . . . . . . 10 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑖 + 1) ∈ (0...𝑀))
2821, 22, 27iccpartxr 45601 . . . . . . . . 9 ((𝜑𝑖 ∈ (0..^𝑀)) → (𝑃‘(𝑖 + 1)) ∈ ℝ*)
2925, 28jca 512 . . . . . . . 8 ((𝜑𝑖 ∈ (0..^𝑀)) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
3029adantrr 715 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑖) ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ*))
31 elico1 13307 . . . . . . 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 13588 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ (0...𝑀))
3635adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → 𝑗 ∈ (0...𝑀))
3733, 34, 36iccpartxr 45601 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃𝑗) ∈ ℝ*)
38 fzofzp1 13669 . . . . . . . . . . 11 (𝑗 ∈ (0..^𝑀) → (𝑗 + 1) ∈ (0...𝑀))
3938adantl 482 . . . . . . . . . 10 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑗 + 1) ∈ (0...𝑀))
4033, 34, 39iccpartxr 45601 . . . . . . . . 9 ((𝜑𝑗 ∈ (0..^𝑀)) → (𝑃‘(𝑗 + 1)) ∈ ℝ*)
4137, 40jca 512 . . . . . . . 8 ((𝜑𝑗 ∈ (0..^𝑀)) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
4241adantrl 714 . . . . . . 7 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑃𝑗) ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ*))
43 elico1 13307 . . . . . . 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 631 . . . . 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 13572 . . . . . . . . . 10 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℤ)
4746zred 12607 . . . . . . . . 9 (𝑖 ∈ (0..^𝑀) → 𝑖 ∈ ℝ)
48 elfzoelz 13572 . . . . . . . . . 10 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
4948zred 12607 . . . . . . . . 9 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
5047, 49anim12i 613 . . . . . . . 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 11239 . . . . . . 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 45617 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑖 < 𝑗 → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))))
5554imp31 418 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑖 < 𝑗) → (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗))
56 simpl 483 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → 𝑋 ∈ ℝ*)
5728adantrr 715 . . . . . . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . . . . . . . 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 45535 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑖 + 1)) ∈ ℝ* ∧ (𝑃𝑗) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6256, 58, 60, 61syl3anc 1371 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋))
6362pm2.21d 121 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑖 + 1)) ∧ (𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) ∧ (𝑃𝑗) ≤ 𝑋) → 𝑖 = 𝑗))
64633expd 1353 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑖 + 1)) → ((𝑃‘(𝑖 + 1)) ≤ (𝑃𝑗) → ((𝑃𝑗) ≤ 𝑋𝑖 = 𝑗))))
6564ex 413 . . . . . . . . . . . . . . . . . 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 422 . . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 1135 . . . . . . . . . . 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 407 . . . . . . . . . 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 591 . . . . . . . 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 414 . . . . . . 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 45617 . . . . . . . . . . 11 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
7978ancomsd 466 . . . . . . . . . 10 (𝜑 → ((𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)) → (𝑗 < 𝑖 → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))))
8079imp31 418 . . . . . . . . 9 (((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) ∧ 𝑗 < 𝑖) → (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖))
8140adantrl 714 . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . . 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 45535 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ∈ ℝ* ∧ (𝑃‘(𝑗 + 1)) ∈ ℝ* ∧ (𝑃𝑖) ∈ ℝ*) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8656, 82, 84, 85syl3anc 1371 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ¬ (𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋))
8786pm2.21d 121 . . . . . . . . . . . . . . . . . . 19 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → ((𝑋 < (𝑃‘(𝑗 + 1)) ∧ (𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) ∧ (𝑃𝑖) ≤ 𝑋) → 𝑖 = 𝑗))
88873expd 1353 . . . . . . . . . . . . . . . . . 18 ((𝑋 ∈ ℝ* ∧ (𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀)))) → (𝑋 < (𝑃‘(𝑗 + 1)) → ((𝑃‘(𝑗 + 1)) ≤ (𝑃𝑖) → ((𝑃𝑖) ≤ 𝑋𝑖 = 𝑗))))
8988ex 413 . . . . . . . . . . . . . . . . 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 422 . . . . . . . . . . . . . . 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 1131 . . . . . . . . . . . . 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 1134 . . . . . . . . . . 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 407 . . . . . . . . . 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 591 . . . . . . . 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 414 . . . . . . 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 1427 . . . . . 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 239 . . . 4 ((𝜑 ∧ (𝑖 ∈ (0..^𝑀) ∧ 𝑗 ∈ (0..^𝑀))) → ((𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ∧ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))) → 𝑖 = 𝑗))
103102ralrimivva 3197 . . 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 6842 . . . . 5 (𝑖 = 𝑗 → (𝑃𝑖) = (𝑃𝑗))
106 fvoveq1 7380 . . . . 5 (𝑖 = 𝑗 → (𝑃‘(𝑖 + 1)) = (𝑃‘(𝑗 + 1)))
107105, 106oveq12d 7375 . . . 4 (𝑖 = 𝑗 → ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) = ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1))))
108107eleq2d 2823 . . 3 (𝑖 = 𝑗 → (𝑋 ∈ ((𝑃𝑖)[,)(𝑃‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑃𝑗)[,)(𝑃‘(𝑗 + 1)))))
109108reu4 3689 . 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 583 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 205  wa 396  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wrex 3073  ∃!wreu 3351   class class class wbr 5105  cfv 6496  (class class class)co 7357  cr 11050  0cc0 11051  1c1 11052   + caddc 11054  *cxr 11188   < clt 11189  cle 11190  cn 12153  [,)cico 13266  ...cfz 13424  ..^cfzo 13567  RePartciccp 45595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-ico 13270  df-fz 13425  df-fzo 13568  df-iccp 45596
This theorem is referenced by:  iccpartdisj  45619
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