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Theorem iccpartgt 41688
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 11389 . . . . . . . 8 (𝜑𝑀 ∈ ℕ0)
3 elnn0uz 11763 . . . . . . . 8 (𝑀 ∈ ℕ0𝑀 ∈ (ℤ‘0))
42, 3sylib 208 . . . . . . 7 (𝜑𝑀 ∈ (ℤ‘0))
5 fzpred 12427 . . . . . . 7 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
64, 5syl 17 . . . . . 6 (𝜑 → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
7 0p1e1 11170 . . . . . . . . 9 (0 + 1) = 1
87oveq1i 6700 . . . . . . . 8 ((0 + 1)...𝑀) = (1...𝑀)
98a1i 11 . . . . . . 7 (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
109uneq2d 3800 . . . . . 6 (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
116, 10eqtrd 2685 . . . . 5 (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
1211eleq2d 2716 . . . 4 (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13 elun 3786 . . . . . . 7 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14 velsn 4226 . . . . . . . 8 (𝑖 ∈ {0} ↔ 𝑖 = 0)
1514orbi1i 541 . . . . . . 7 ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
1613, 15bitri 264 . . . . . 6 (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17 fzisfzounsn 12620 . . . . . . . . . . 11 (𝑀 ∈ (ℤ‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
184, 17syl 17 . . . . . . . . . 10 (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
1918eleq2d 2716 . . . . . . . . 9 (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20 elun 3786 . . . . . . . . . 10 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21 velsn 4226 . . . . . . . . . . 11 (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
2221orbi2i 540 . . . . . . . . . 10 ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2320, 22bitri 264 . . . . . . . . 9 (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
2419, 23syl6bb 276 . . . . . . . 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 10630 . . . . . . . . . . . . . . . . . . 19 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28 fzo1fzo0n0 12558 . . . . . . . . . . . . . . . . . . 19 (𝑗 ∈ (1..^𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∧ 𝑗 ≠ 0))
2925, 27, 28sylanbrc 699 . . . . . . . . . . . . . . . . . 18 ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (1..^𝑀))
30 iccpartgtprec.p . . . . . . . . . . . . . . . . . . 19 (𝜑𝑃 ∈ (RePart‘𝑀))
311, 30iccpartigtl 41684 . . . . . . . . . . . . . . . . . 18 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃𝑘))
32 fveq2 6229 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑗 → (𝑃𝑘) = (𝑃𝑗))
3332breq2d 4697 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃𝑘) ↔ (𝑃‘0) < (𝑃𝑗)))
3433rspcv 3336 . . . . . . . . . . . . . . . . . 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 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 4688 . . . . . . . . . . . . . . . 16 (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39 fveq2 6229 . . . . . . . . . . . . . . . . 17 (𝑖 = 0 → (𝑃𝑖) = (𝑃‘0))
4039breq1d 4695 . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . 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 41685 . . . . . . . . . . . . . . 15 (𝜑 → (𝑃‘0) < (𝑃𝑀))
46 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑗 = 𝑀 → (𝑃𝑗) = (𝑃𝑀))
4739, 46breqan12rd 4702 . . . . . . . . . . . . . . 15 ((𝑗 = 𝑀𝑖 = 0) → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃‘0) < (𝑃𝑀)))
4845, 47syl5ibr 236 . . . . . . . . . . . . . 14 ((𝑗 = 𝑀𝑖 = 0) → (𝜑 → (𝑃𝑖) < (𝑃𝑗)))
4948a1dd 50 . . . . . . . . . . . . 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 12380 . . . . . . . . . . . . . . . . 17 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
5453ad3antlr 767 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
5553peano2zd 11523 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
5655ad2antlr 763 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57 elfzoelz 12509 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
5857ad2antrr 762 . . . . . . . . . . . . . . . . . . 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 590 . . . . . . . . . . . . . . . . . . . . . 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 11465 . . . . . . . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
6556, 58, 643jca 1261 . . . . . . . . . . . . . . . . . 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 11731 . . . . . . . . . . . . . . . . 17 (𝑗 ∈ (ℤ‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
6866, 67sylibr 224 . . . . . . . . . . . . . . . 16 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ‘(𝑖 + 1)))
691ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
7030ad2antlr 763 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71 1zzd 11446 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72 elfzelz 12380 . . . . . . . . . . . . . . . . . . . 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 12382 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75 elfzle1 12382 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → 𝑖𝑘)
76 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77 elfzel1 12379 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
7877zred 11520 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
7972zred 11520 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80 letr 10169 . . . . . . . . . . . . . . . . . . . . . . . 24 ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8176, 78, 79, 80syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖𝑖𝑘) → 1 ≤ 𝑘))
8275, 81mpan2d 710 . . . . . . . . . . . . . . . . . . . . . 22 (𝑘 ∈ (𝑖...𝑗) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
8374, 82syl5com 31 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 ∈ (1...𝑀) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
8483ad3antlr 767 . . . . . . . . . . . . . . . . . . . 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 11731 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ (ℤ‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
8771, 73, 85, 86syl3anbrc 1265 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ‘1))
88 elfzel2 12378 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
8988ad2antlr 763 . . . . . . . . . . . . . . . . . . 19 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
9089ad2antrr 762 . . . . . . . . . . . . . . . . . 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 11520 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
9392ad4antr 769 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
9469nnred 11073 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95 elfzle2 12383 . . . . . . . . . . . . . . . . . . . 20 (𝑘 ∈ (𝑖...𝑗) → 𝑘𝑗)
9695adantl 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘𝑗)
97 elfzolt2 12518 . . . . . . . . . . . . . . . . . . . 20 (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
9897ad4antr 769 . . . . . . . . . . . . . . . . . . 19 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
9991, 93, 94, 96, 98lelttrd 10233 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100 elfzo2 12512 . . . . . . . . . . . . . . . . . 18 (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
10187, 90, 99, 100syl3anbrc 1265 . . . . . . . . . . . . . . . . 17 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
10269, 70, 101iccpartipre 41682 . . . . . . . . . . . . . . . 16 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃𝑘) ∈ ℝ)
1031ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
10430ad2antlr 763 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
10557ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106 fzoval 12510 . . . . . . . . . . . . . . . . . . . . 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 12551 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → 𝑗𝑀)
109 0le1 10589 . . . . . . . . . . . . . . . . . . . . . . . . . 26 0 ≤ 1
110 0red 10079 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111 1red 10093 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
11253zred 11520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113 letr 10169 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114110, 111, 112, 113syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115109, 114mpani 712 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑖 ∈ (1...𝑀) → (1 ≤ 𝑖 → 0 ≤ 𝑖))
11674, 115mpd 15 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
117108, 116anim12ci 590 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (0 ≤ 𝑖𝑗𝑀))
118117adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖𝑗𝑀))
119 0zd 11427 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120 elfzoel2 12508 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121119, 120jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122121ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123 ssfzo12bi 12603 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
12461, 122, 59, 123syl3anc 1366 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖𝑗𝑀)))
125118, 124mpbird 247 . . . . . . . . . . . . . . . . . . . . 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 3673 . . . . . . . . . . . . . . . . . . 19 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128127sselda 3636 . . . . . . . . . . . . . . . . . 18 (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129 iccpartimp 41678 . . . . . . . . . . . . . . . . . 18 ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ*𝑚 (0...𝑀)) ∧ (𝑃𝑘) < (𝑃‘(𝑘 + 1))))
130103, 104, 128, 129syl3anc 1366 . . . . . . . . . . . . . . . . 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 41666 . . . . . . . . . . . . . . 15 ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃𝑖) < (𝑃𝑗))
133132exp31 629 . . . . . . . . . . . . . 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 449 . . . . . . . . . . . 12 (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
136 elfzuz 12376 . . . . . . . . . . . . . . . . . . . . 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 12512 . . . . . . . . . . . . . . . . . . . 20 (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141137, 138, 139, 140syl3anbrc 1265 . . . . . . . . . . . . . . . . . . 19 ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
1421, 30iccpartiltu 41683 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃𝑘) < (𝑃𝑀))
143 fveq2 6229 . . . . . . . . . . . . . . . . . . . . 21 (𝑘 = 𝑖 → (𝑃𝑘) = (𝑃𝑖))
144143breq1d 4695 . . . . . . . . . . . . . . . . . . . 20 (𝑘 = 𝑖 → ((𝑃𝑘) < (𝑃𝑀) ↔ (𝑃𝑖) < (𝑃𝑀)))
145144rspcv 3336 . . . . . . . . . . . . . . . . . . 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 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 4689 . . . . . . . . . . . . . . 15 (𝑗 = 𝑀 → (𝑖 < 𝑗𝑖 < 𝑀))
152151anbi2d 740 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) ↔ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀)))
15346breq2d 4697 . . . . . . . . . . . . . 14 (𝑗 = 𝑀 → ((𝑃𝑖) < (𝑃𝑗) ↔ (𝑃𝑖) < (𝑃𝑀)))
154150, 152, 1533imtr4d 283 . . . . . . . . . . . . 13 (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃𝑖) < (𝑃𝑗)))
155154exp4c 635 . . . . . . . . . . . 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 88 . . . . . . . 8 (𝜑 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))))
16024, 159sylbid 230 . . . . . . 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 207 . . . . 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 230 . . 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 3000 1 (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃𝑖) < (𝑃𝑗)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  cun 3605  wss 3607  {csn 4210   class class class wbr 4685  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cmin 10304  cn 11058  0cn0 11330  cz 11415  cuz 11725  ...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-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-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-fz 12365  df-fzo 12505  df-iccp 41675
This theorem is referenced by:  icceuelpartlem  41696  iccpartnel  41699
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