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Theorem iccpartgt 43937
 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 11947 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3 elnn0uz 12275 . . . . . . . 8 (𝑀 ∈ ℕ0𝑀 ∈ (ℤ‘0))
42, 3sylib 221 . . . . . . 7 (𝜑𝑀 ∈ (ℤ‘0))
5 fzpred 12954 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
64, 5syl 17 . . . . . 6 (𝜑 → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
7 0p1e1 11751 . . . . . . . . 9 (0 + 1) = 1
87oveq1i 7149 . . . . . . . 8 ((0 + 1)...𝑀) = (1...𝑀)
98a1i 11 . . . . . . 7 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
109uneq2d 4093 . . . . . 6 (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
116, 10eqtrd 2836 . . . . 5 (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
1211eleq2d 2878 . . . 4 (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13 elun 4079 . . . . . . 7 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14 velsn 4544 . . . . . . . 8 (𝑖 ∈ {0} ↔ 𝑖 = 0)
1514orbi1i 911 . . . . . . 7 ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
1613, 15bitri 278 . . . . . 6 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17 fzisfzounsn 13148 . . . . . . . . . . 11 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
184, 17syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
1918eleq2d 2878 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20 elun 4079 . . . . . . . . . 10 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21 velsn 4544 . . . . . . . . . . 11 (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
2221orbi2i 910 . . . . . . . . . 10 ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2320, 22bitri 278 . . . . . . . . 9 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2419, 23syl6bb 290 . . . . . . . 8 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀)))
25 simpl 486 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (0..^𝑀))
26 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 0 < 𝑗)
2726gt0ne0d 11197 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28 fzo1fzo0n0 13087 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1..^𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∧ 𝑗 ≠ 0))
2925, 27, 28sylanbrc 586 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (1..^𝑀))
30 iccpartgtprec.p . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃 ∈ (RePart‘𝑀))
311, 30iccpartigtl 43933 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑘))
32 fveq2 6649 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (𝑃𝑘) = (𝑃𝑗))
3332breq2d 5045 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃𝑘) ↔ (𝑃‘0) < (𝑃𝑗)))
3433rspcv 3569 . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑗 ∈ (0..^𝑀) → (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗))))
3736impcom 411 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗)))
38 breq1 5036 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39 fveq2 6649 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑃𝑖) = (𝑃‘0))
4039breq1d 5043 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑗)))
4138, 40imbi12d 348 . . . . . . . . . . . . . . 15 (𝑖 = 0 → ((𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)) ↔ (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗))))
4237, 41syl5ibr 249 . . . . . . . . . . . . . 14 (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
4342expd 419 . . . . . . . . . . . . 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 43934 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘0) < (𝑃𝑀))
46 fveq2 6649 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (𝑃𝑗) = (𝑃𝑀))
4739, 46breqan12rd 5050 . . . . . . . . . . . . . . 15 ((𝑗 = 𝑀𝑖 = 0) → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑀)))
4845, 47syl5ibr 249 . . . . . . . . . . . . . 14 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑃𝑖) < (𝑃𝑗)))
4948a1dd 50 . . . . . . . . . . . . 13 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
5049ex 416 . . . . . . . . . . . 12 (𝑗 = 𝑀 → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
5144, 50jaoi 854 . . . . . . . . . . 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 12906 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
5453ad3antlr 730 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
5553peano2zd 12082 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
5655ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57 elfzoelz 13037 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
5857ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
59 simpr 488 . . . . . . . . . . . . . . . . . . . 20 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
6057, 53anim12ci 616 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
6160adantr 484 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
62 zltp1le 12024 . . . . . . . . . . . . . . . . . . . . 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 235 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
6556, 58, 643jca 1125 . . . . . . . . . . . . . . . . . 18 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6665adantr 484 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
67 eluz2 12241 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6866, 67sylibr 237 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
691ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
7030ad2antlr 726 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71 1zzd 12005 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72 elfzelz 12906 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℤ)
7372adantl 485 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℤ)
74 elfzle1 12909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75 elfzle1 12909 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → 𝑖𝑘)
76 1red 10635 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77 elfzel1 12905 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
7877zred 12079 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
7972zred 12079 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80 letr 10727 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8176, 78, 79, 80syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8275, 81mpan2d 693 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑖...𝑗) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
8374, 82syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8483ad3antlr 730 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8584imp 410 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ≤ 𝑘)
86 eluz2 12241 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
8771, 73, 85, 86syl3anbrc 1340 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ‘1))
88 elfzel2 12904 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
8988ad2antlr 726 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
9089ad2antrr 725 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℤ)
9179adantl 485 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℝ)
9257zred 12079 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
9392ad4antr 731 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
9469nnred 11644 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95 elfzle2 12910 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘𝑗)
9695adantl 485 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘𝑗)
97 elfzolt2 13046 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
9897ad4antr 731 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
9991, 93, 94, 96, 98lelttrd 10791 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100 elfzo2 13040 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
10187, 90, 99, 100syl3anbrc 1340 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
10269, 70, 101iccpartipre 43931 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃𝑘) ∈ ℝ)
1031ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
10430ad2antlr 726 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
10557ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106 fzoval 13038 . . . . . . . . . . . . . . . . . . . . 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 13080 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → 𝑗𝑀)
109 0le1 11156 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ≤ 1
110 0red 10637 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111 1red 10635 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
11253zred 12079 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113 letr 10727 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114110, 111, 112, 113syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115109, 114mpani 695 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (1...𝑀) → (1 ≤ 𝑖 → 0 ≤ 𝑖))
11674, 115mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
117108, 116anim12ci 616 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (0 ≤ 𝑖𝑗𝑀))
118117adantr 484 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖𝑗𝑀))
119 0zd 11985 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120 elfzoel2 13036 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121119, 120jca 515 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122121ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123 ssfzo12bi 13131 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
12461, 122, 59, 123syl3anc 1368 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
125118, 124mpbird 260 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖..^𝑗) ⊆ (0..^𝑀))
126125adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) ⊆ (0..^𝑀))
127107, 126eqsstrrd 3957 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128127sselda 3918 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129 iccpartimp 43927 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
130103, 104, 128, 129syl3anc 1368 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃 ∈ (ℝ*m (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
131130simprd 499 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃𝑘) < (𝑃‘(𝑘 + 1)))
13254, 68, 102, 131smonoord 43881 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃𝑖) < (𝑃𝑗))
133132exp31 423 . . . . . . . . . . . . . 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 416 . . . . . . . . . . . 12 (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
136 elfzuz 12902 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (ℤ‘1))
137136adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (ℤ‘1))
13888adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
139 simpr 488 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
140 elfzo2 13040 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141137, 138, 139, 140syl3anbrc 1340 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
1421, 30iccpartiltu 43932 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃𝑘) < (𝑃𝑀))
143 fveq2 6649 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
144143breq1d 5043 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → ((𝑃𝑘) < (𝑃𝑀) ↔ (𝑃𝑖) < (𝑃𝑀)))
145144rspcv 3569 . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑖 < 𝑀 → (𝑃𝑖) < (𝑃𝑀))))
148147impcom 411 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (1...𝑀) ∧ 𝜑) → (𝑖 < 𝑀 → (𝑃𝑖) < (𝑃𝑀)))
149148imp 410 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀))
150149a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀)))
151 breq2 5037 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (𝑖 < 𝑗𝑖 < 𝑀))
152151anbi2d 631 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) ↔ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀)))
15346breq2d 5045 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃𝑖) < (𝑃𝑀)))
154150, 152, 1533imtr4d 297 . . . . . . . . . . . . 13 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃𝑖) < (𝑃𝑗)))
155154exp4c 436 . . . . . . . . . . . 12 (𝑗 = 𝑀 → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
156135, 155jaoi 854 . . . . . . . . . . 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 854 . . . . . . . . 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 243 . . . . . . 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 220 . . . . 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 243 . . 3 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
165164imp32 422 . 2 ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
166165ralrimivva 3159 1 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  ∀wral 3109   ∪ cun 3882   ⊆ wss 3884  {csn 4528   class class class wbr 5033  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393  ℝcr 10529  0cc0 10530  1c1 10531   + caddc 10533  ℝ*cxr 10667   < clt 10668   ≤ cle 10669   − cmin 10863  ℕcn 11629  ℕ0cn0 11889  ℤcz 11973  ℤ≥cuz 12235  ...cfz 12889  ..^cfzo 13032  RePartciccp 43923 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-iccp 43924 This theorem is referenced by:  icceuelpartlem  43945  iccpartnel  43948
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