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Theorem iccpartgt 42029
Description: If there is a partition, then all intermediate points and the bounds are strictly ordered. (Contributed by AV, 18-Jul-2020.)
Hypotheses
Ref Expression
iccpartgtprec.m (𝜑𝑀 ∈ ℕ)
iccpartgtprec.p (𝜑𝑃 ∈ (RePart‘𝑀))
Assertion
Ref Expression
iccpartgt (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
Distinct variable groups:   𝑖,𝑀   𝑃,𝑖   𝜑,𝑖   𝑗,𝑀   𝑃,𝑗,𝑖   𝜑,𝑗

Proof of Theorem iccpartgt
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 iccpartgtprec.m . . . . . . . . 9 (𝜑𝑀 ∈ ℕ)
21nnnn0d 11598 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3 elnn0uz 11925 . . . . . . . 8 (𝑀 ∈ ℕ0𝑀 ∈ (ℤ‘0))
42, 3sylib 209 . . . . . . 7 (𝜑𝑀 ∈ (ℤ‘0))
5 fzpred 12596 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
64, 5syl 17 . . . . . 6 (𝜑 → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
7 0p1e1 11401 . . . . . . . . 9 (0 + 1) = 1
87oveq1i 6852 . . . . . . . 8 ((0 + 1)...𝑀) = (1...𝑀)
98a1i 11 . . . . . . 7 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
109uneq2d 3929 . . . . . 6 (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
116, 10eqtrd 2799 . . . . 5 (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
1211eleq2d 2830 . . . 4 (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13 elun 3915 . . . . . . 7 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14 velsn 4350 . . . . . . . 8 (𝑖 ∈ {0} ↔ 𝑖 = 0)
1514orbi1i 937 . . . . . . 7 ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
1613, 15bitri 266 . . . . . 6 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17 fzisfzounsn 12788 . . . . . . . . . . 11 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
184, 17syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
1918eleq2d 2830 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20 elun 3915 . . . . . . . . . 10 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21 velsn 4350 . . . . . . . . . . 11 (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
2221orbi2i 936 . . . . . . . . . 10 ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2320, 22bitri 266 . . . . . . . . 9 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2419, 23syl6bb 278 . . . . . . . 8 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀)))
25 simpl 474 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (0..^𝑀))
26 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 0 < 𝑗)
2726gt0ne0d 10846 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28 fzo1fzo0n0 12727 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1..^𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∧ 𝑗 ≠ 0))
2925, 27, 28sylanbrc 578 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (1..^𝑀))
30 iccpartgtprec.p . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃 ∈ (RePart‘𝑀))
311, 30iccpartigtl 42025 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑘))
32 fveq2 6375 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (𝑃𝑘) = (𝑃𝑗))
3332breq2d 4821 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃𝑘) ↔ (𝑃‘0) < (𝑃𝑗)))
3433rspcv 3457 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑘) → (𝑃‘0) < (𝑃𝑗)))
3529, 31, 34syl2imc 41 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → (𝑃‘0) < (𝑃𝑗)))
3635expd 404 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑗 ∈ (0..^𝑀) → (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗))))
3736impcom 396 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗)))
38 breq1 4812 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39 fveq2 6375 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑃𝑖) = (𝑃‘0))
4039breq1d 4819 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑗)))
4138, 40imbi12d 335 . . . . . . . . . . . . . . 15 (𝑖 = 0 → ((𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)) ↔ (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗))))
4237, 41syl5ibr 237 . . . . . . . . . . . . . 14 (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
4342expd 404 . . . . . . . . . . . . 13 (𝑖 = 0 → (𝑗 ∈ (0..^𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
4443com12 32 . . . . . . . . . . . 12 (𝑗 ∈ (0..^𝑀) → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
451, 30iccpartlt 42026 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘0) < (𝑃𝑀))
46 fveq2 6375 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (𝑃𝑗) = (𝑃𝑀))
4739, 46breqan12rd 4826 . . . . . . . . . . . . . . 15 ((𝑗 = 𝑀𝑖 = 0) → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑀)))
4845, 47syl5ibr 237 . . . . . . . . . . . . . 14 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑃𝑖) < (𝑃𝑗)))
4948a1dd 50 . . . . . . . . . . . . 13 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
5049ex 401 . . . . . . . . . . . 12 (𝑗 = 𝑀 → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
5144, 50jaoi 883 . . . . . . . . . . 11 ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
5251com12 32 . . . . . . . . . 10 (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
53 elfzelz 12549 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
5453ad3antlr 722 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
5553peano2zd 11732 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
5655ad2antlr 718 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57 elfzoelz 12678 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
5857ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
59 simpr 477 . . . . . . . . . . . . . . . . . . . 20 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
6057, 53anim12ci 607 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
6160adantr 472 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
62 zltp1le 11674 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
6361, 62syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
6459, 63mpbid 223 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
6556, 58, 643jca 1158 . . . . . . . . . . . . . . . . . 18 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6665adantr 472 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
67 eluz2 11892 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6866, 67sylibr 225 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
691ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
7030ad2antlr 718 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71 1zzd 11655 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72 elfzelz 12549 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℤ)
7372adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℤ)
74 elfzle1 12551 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75 elfzle1 12551 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → 𝑖𝑘)
76 1red 10294 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77 elfzel1 12548 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
7877zred 11729 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
7972zred 11729 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80 letr 10385 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8176, 78, 79, 80syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8275, 81mpan2d 685 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑖...𝑗) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
8374, 82syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8483ad3antlr 722 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8584imp 395 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ≤ 𝑘)
86 eluz2 11892 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
8771, 73, 85, 86syl3anbrc 1443 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ‘1))
88 elfzel2 12547 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
8988ad2antlr 718 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
9089ad2antrr 717 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℤ)
9179adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℝ)
9257zred 11729 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
9392ad4antr 724 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
9469nnred 11291 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95 elfzle2 12552 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘𝑗)
9695adantl 473 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘𝑗)
97 elfzolt2 12687 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
9897ad4antr 724 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
9991, 93, 94, 96, 98lelttrd 10449 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100 elfzo2 12681 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
10187, 90, 99, 100syl3anbrc 1443 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
10269, 70, 101iccpartipre 42023 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃𝑘) ∈ ℝ)
1031ad2antlr 718 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
10430ad2antlr 718 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
10557ad3antrrr 721 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106 fzoval 12679 . . . . . . . . . . . . . . . . . . . . 21 (𝑗 ∈ ℤ → (𝑖..^𝑗) = (𝑖...(𝑗 − 1)))
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) = (𝑖...(𝑗 − 1)))
108 elfzo0le 12720 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → 𝑗𝑀)
109 0le1 10805 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ≤ 1
110 0red 10297 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111 1red 10294 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
11253zred 11729 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113 letr 10385 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114110, 111, 112, 113syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115109, 114mpani 687 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (1...𝑀) → (1 ≤ 𝑖 → 0 ≤ 𝑖))
11674, 115mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
117108, 116anim12ci 607 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (0 ≤ 𝑖𝑗𝑀))
118117adantr 472 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖𝑗𝑀))
119 0zd 11636 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120 elfzoel2 12677 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121119, 120jca 507 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122121ad2antrr 717 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123 ssfzo12bi 12771 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
12461, 122, 59, 123syl3anc 1490 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
125118, 124mpbird 248 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖..^𝑗) ⊆ (0..^𝑀))
126125adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) ⊆ (0..^𝑀))
127107, 126eqsstr3d 3800 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128127sselda 3761 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129 iccpartimp 42019 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
130103, 104, 128, 129syl3anc 1490 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
131130simprd 489 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃𝑘) < (𝑃‘(𝑘 + 1)))
13254, 68, 102, 131smonoord 42007 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃𝑖) < (𝑃𝑗))
133132exp31 410 . . . . . . . . . . . . . 14 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝜑 → (𝑃𝑖) < (𝑃𝑗))))
134133com23 86 . . . . . . . . . . . . 13 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
135134ex 401 . . . . . . . . . . . 12 (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
136 elfzuz 12545 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (ℤ‘1))
137136adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (ℤ‘1))
13888adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
139 simpr 477 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
140 elfzo2 12681 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141137, 138, 139, 140syl3anbrc 1443 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
1421, 30iccpartiltu 42024 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃𝑘) < (𝑃𝑀))
143 fveq2 6375 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
144143breq1d 4819 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → ((𝑃𝑘) < (𝑃𝑀) ↔ (𝑃𝑖) < (𝑃𝑀)))
145144rspcv 3457 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃𝑘) < (𝑃𝑀) → (𝑃𝑖) < (𝑃𝑀)))
146141, 142, 145syl2imc 41 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀)))
147146expd 404 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑖 < 𝑀 → (𝑃𝑖) < (𝑃𝑀))))
148147impcom 396 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (1...𝑀) ∧ 𝜑) → (𝑖 < 𝑀 → (𝑃𝑖) < (𝑃𝑀)))
149148imp 395 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀))
150149a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀)))
151 breq2 4813 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (𝑖 < 𝑗𝑖 < 𝑀))
152151anbi2d 622 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) ↔ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀)))
15346breq2d 4821 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃𝑖) < (𝑃𝑀)))
154150, 152, 1533imtr4d 285 . . . . . . . . . . . . 13 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃𝑖) < (𝑃𝑗)))
155154exp4c 423 . . . . . . . . . . . 12 (𝑗 = 𝑀 → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
156135, 155jaoi 883 . . . . . . . . . . 11 ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
157156com12 32 . . . . . . . . . 10 (𝑖 ∈ (1...𝑀) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
15852, 157jaoi 883 . . . . . . . . 9 ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
159158com13 88 . . . . . . . 8 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
16024, 159sylbid 231 . . . . . . 7 (𝜑 → (𝑗 ∈ (0...𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
161160com3r 87 . . . . . 6 ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
16216, 161sylbi 208 . . . . 5 (𝑖 ∈ ({0} ∪ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
163162com12 32 . . . 4 (𝜑 → (𝑖 ∈ ({0} ∪ (1...𝑀)) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
16412, 163sylbid 231 . . 3 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
165164imp32 409 . 2 ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
166165ralrimivva 3118 1 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wo 873  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  cun 3730  wss 3732  {csn 4334   class class class wbr 4809  cfv 6068  (class class class)co 6842  𝑚 cmap 8060  cr 10188  0cc0 10189  1c1 10190   + caddc 10192  *cxr 10327   < clt 10328  cle 10329  cmin 10520  cn 11274  0cn0 11538  cz 11624  cuz 11886  ...cfz 12533  ..^cfzo 12673  RePartciccp 42015
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 2069  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 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  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-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-iccp 42016
This theorem is referenced by:  icceuelpartlem  42037  iccpartnel  42040
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