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Theorem iccpartgt 39965
 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 11228 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3 elnn0uz 11601 . . . . . . . 8 (𝑀 ∈ ℕ0𝑀 ∈ (ℤ‘0))
42, 3sylib 207 . . . . . . 7 (𝜑𝑀 ∈ (ℤ‘0))
5 fzpred 12259 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
64, 5syl 17 . . . . . 6 (𝜑 → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
7 0p1e1 11009 . . . . . . . . 9 (0 + 1) = 1
87oveq1i 6559 . . . . . . . 8 ((0 + 1)...𝑀) = (1...𝑀)
98a1i 11 . . . . . . 7 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
109uneq2d 3729 . . . . . 6 (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
116, 10eqtrd 2644 . . . . 5 (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
1211eleq2d 2673 . . . 4 (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13 elun 3715 . . . . . . 7 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14 velsn 4141 . . . . . . . 8 (𝑖 ∈ {0} ↔ 𝑖 = 0)
1514orbi1i 541 . . . . . . 7 ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
1613, 15bitri 263 . . . . . 6 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17 fzisfzounsn 12445 . . . . . . . . . . 11 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
184, 17syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
1918eleq2d 2673 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20 elun 3715 . . . . . . . . . 10 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21 velsn 4141 . . . . . . . . . . 11 (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
2221orbi2i 540 . . . . . . . . . 10 ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2320, 22bitri 263 . . . . . . . . 9 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2419, 23syl6bb 275 . . . . . . . 8 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀)))
25 simpl 472 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (0..^𝑀))
26 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 0 < 𝑗)
2726gt0ne0d 10471 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28 fzo1fzo0n0 12386 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1..^𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∧ 𝑗 ≠ 0))
2925, 27, 28sylanbrc 695 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (1..^𝑀))
30 iccpartgtprec.p . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃 ∈ (RePart‘𝑀))
311, 30iccpartigtl 39961 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑘))
32 fveq2 6103 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (𝑃𝑘) = (𝑃𝑗))
3332breq2d 4595 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃𝑘) ↔ (𝑃‘0) < (𝑃𝑗)))
3433rspcv 3278 . . . . . . . . . . . . . . . . . 18 (𝑗 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑘) → (𝑃‘0) < (𝑃𝑗)))
3529, 31, 34syl2imc 40 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → (𝑃‘0) < (𝑃𝑗)))
3635expd 451 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑗 ∈ (0..^𝑀) → (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗))))
3736impcom 445 . . . . . . . . . . . . . . 15 ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗)))
38 breq1 4586 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39 fveq2 6103 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑃𝑖) = (𝑃‘0))
4039breq1d 4593 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑗)))
4138, 40imbi12d 333 . . . . . . . . . . . . . . 15 (𝑖 = 0 → ((𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)) ↔ (0 < 𝑗 → (𝑃‘0) < (𝑃𝑗))))
4237, 41syl5ibr 235 . . . . . . . . . . . . . 14 (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
4342expd 451 . . . . . . . . . . . . 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 39962 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘0) < (𝑃𝑀))
46 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (𝑃𝑗) = (𝑃𝑀))
4739, 46breqan12rd 4600 . . . . . . . . . . . . . . 15 ((𝑗 = 𝑀𝑖 = 0) → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑀)))
4845, 47syl5ibr 235 . . . . . . . . . . . . . 14 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑃𝑖) < (𝑃𝑗)))
4948a1dd 48 . . . . . . . . . . . . 13 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
5049ex 449 . . . . . . . . . . . 12 (𝑗 = 𝑀 → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
5144, 50jaoi 393 . . . . . . . . . . 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 12213 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
5453ad3antlr 763 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
5553peano2zd 11361 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
5655ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57 elfzoelz 12339 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
5857ad2antrr 758 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
59 simpr 476 . . . . . . . . . . . . . . . . . . . 20 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
6057, 53anim12ci 589 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
6160adantr 480 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
62 zltp1le 11304 . . . . . . . . . . . . . . . . . . . . 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 221 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
6556, 58, 643jca 1235 . . . . . . . . . . . . . . . . . 18 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6665adantr 480 . . . . . . . . . . . . . . . . 17 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
67 eluz2 11569 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6866, 67sylibr 223 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
691ad2antlr 759 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
7030ad2antlr 759 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71 1zzd 11285 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72 elfzelz 12213 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℤ)
7372adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℤ)
74 elfzle1 12215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75 elfzle1 12215 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → 𝑖𝑘)
76 1red 9934 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77 elfzel1 12212 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
7877zred 11358 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
7972zred 11358 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80 letr 10010 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8176, 78, 79, 80syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8275, 81mpan2d 706 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑖...𝑗) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
8374, 82syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8483ad3antlr 763 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8584imp 444 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ≤ 𝑘)
86 eluz2 11569 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
8771, 73, 85, 86syl3anbrc 1239 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ‘1))
88 elfzel2 12211 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
8988ad2antlr 759 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
9089ad2antrr 758 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℤ)
9179adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℝ)
9257zred 11358 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
9392ad4antr 764 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
9469nnred 10912 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95 elfzle2 12216 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘𝑗)
9695adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘𝑗)
97 elfzolt2 12348 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
9897ad4antr 764 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
9991, 93, 94, 96, 98lelttrd 10074 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100 elfzo2 12342 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
10187, 90, 99, 100syl3anbrc 1239 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
10269, 70, 101iccpartipre 39959 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃𝑘) ∈ ℝ)
1031ad2antlr 759 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
10430ad2antlr 759 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
10557ad3antrrr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106 fzoval 12340 . . . . . . . . . . . . . . . . . . . . 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 12379 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → 𝑗𝑀)
109 0le1 10430 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ≤ 1
110 0red 9920 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111 1red 9934 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
11253zred 11358 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113 letr 10010 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114110, 111, 112, 113syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115109, 114mpani 708 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (1...𝑀) → (1 ≤ 𝑖 → 0 ≤ 𝑖))
11674, 115mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
117108, 116anim12ci 589 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (0 ≤ 𝑖𝑗𝑀))
118117adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖𝑗𝑀))
119 0zd 11266 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120 elfzoel2 12338 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121119, 120jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122121ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123 ssfzo12bi 12429 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
12461, 122, 59, 123syl3anc 1318 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
125118, 124mpbird 246 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖..^𝑗) ⊆ (0..^𝑀))
126125adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) ⊆ (0..^𝑀))
127107, 126eqsstr3d 3603 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128127sselda 3568 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129 iccpartimp 39955 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
130103, 104, 128, 129syl3anc 1318 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
131130simprd 478 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃𝑘) < (𝑃‘(𝑘 + 1)))
13254, 68, 102, 131smonoord 39944 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃𝑖) < (𝑃𝑗))
133132exp31 628 . . . . . . . . . . . . . 14 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝜑 → (𝑃𝑖) < (𝑃𝑗))))
134133com23 84 . . . . . . . . . . . . 13 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗))))
135134ex 449 . . . . . . . . . . . 12 (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
136 elfzuz 12209 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (ℤ‘1))
137136adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (ℤ‘1))
13888adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
139 simpr 476 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
140 elfzo2 12342 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141137, 138, 139, 140syl3anbrc 1239 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
1421, 30iccpartiltu 39960 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃𝑘) < (𝑃𝑀))
143 fveq2 6103 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
144143breq1d 4593 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → ((𝑃𝑘) < (𝑃𝑀) ↔ (𝑃𝑖) < (𝑃𝑀)))
145144rspcv 3278 . . . . . . . . . . . . . . . . . . 19 (𝑖 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃𝑘) < (𝑃𝑀) → (𝑃𝑖) < (𝑃𝑀)))
146141, 142, 145syl2imc 40 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀)))
147146expd 451 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑖 < 𝑀 → (𝑃𝑖) < (𝑃𝑀))))
148147impcom 445 . . . . . . . . . . . . . . . 16 ((𝑖 ∈ (1...𝑀) ∧ 𝜑) → (𝑖 < 𝑀 → (𝑃𝑖) < (𝑃𝑀)))
149148imp 444 . . . . . . . . . . . . . . 15 (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀))
150149a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃𝑖) < (𝑃𝑀)))
151 breq2 4587 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (𝑖 < 𝑗𝑖 < 𝑀))
152151anbi2d 736 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) ↔ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀)))
15346breq2d 4595 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃𝑖) < (𝑃𝑀)))
154150, 152, 1533imtr4d 282 . . . . . . . . . . . . 13 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃𝑖) < (𝑃𝑗)))
155154exp4c 634 . . . . . . . . . . . 12 (𝑗 = 𝑀 → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
156135, 155jaoi 393 . . . . . . . . . . 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 393 . . . . . . . . 9 ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
159158com13 86 . . . . . . . 8 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
16024, 159sylbid 229 . . . . . . 7 (𝜑 → (𝑗 ∈ (0...𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
161160com3r 85 . . . . . 6 ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
16216, 161sylbi 206 . . . . 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 229 . . 3 (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
165164imp32 448 . 2 ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
166165ralrimivva 2954 1 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ∪ cun 3538   ⊆ wss 3540  {csn 4125   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549   ↑𝑚 cmap 7744  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145  ℕcn 10897  ℕ0cn0 11169  ℤcz 11254  ℤ≥cuz 11563  ...cfz 12197  ..^cfzo 12334  RePartciccp 39951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-iccp 39952 This theorem is referenced by:  icceuelpartlem  39973  iccpartnel  39976
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