Theorem iccpartgt 48648

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	⊢ (𝜑 → 𝑀 ∈ ℕ)
iccpartgtprec.p	⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))

Assertion

Ref	Expression
iccpartgt	⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))

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.

Step	Hyp	Ref	Expression
1		iccpartgtprec.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2	1	nnnn0d 12677	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
3		elnn0uz 13016	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0))
4	2, 3	sylib 221	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
5		fzpred 13716	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
6	4, 5	syl 18	. . . . . 6 ⊢ (𝜑 → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
7		0p1e1 12473	. . . . . . . . 9 ⊢ (0 + 1) = 1
8	7	oveq1i 7437	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
10	9	uneq2d 4169	. . . . . 6 ⊢ (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
11	6, 10	eqtrd 2798	. . . . 5 ⊢ (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
12	11	eleq2d 2849	. . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13		elun 4154	. . . . . . 7 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14		velsn 4599	. . . . . . . 8 ⊢ (𝑖 ∈ {0} ↔ 𝑖 = 0)
15	14	orbi1i 927	. . . . . . 7 ⊢ ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
16	13, 15	bitri 278	. . . . . 6 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17		fzisfzounsn 13925	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
18	4, 17	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
19	18	eleq2d 2849	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20		elun 4154	. . . . . . . . . 10 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21		velsn 4599	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
22	21	orbi2i 926	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
23	20, 22	bitri 278	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
24	19, 23	bitrdi 290	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀)))
25		simpl 488	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (0..^𝑀))
26		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 0 < 𝑗)
27	26	gt0ne0d 11890	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28		fzo1fzo0n0 13860	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (1..^𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∧ 𝑗 ≠ 0))
29	25, 27, 28	sylanbrc 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (1..^𝑀))
30		iccpartgtprec.p	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
31	1, 30	iccpartigtl 48644	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘))
32		fveq2 6893	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → (𝑃‘𝑘) = (𝑃‘𝑗))
33	32	breq2d 5117	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃‘𝑘) ↔ (𝑃‘0) < (𝑃‘𝑗)))
34	33	rspcv 3575	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘) → (𝑃‘0) < (𝑃‘𝑗)))
35	29, 31, 34	syl2imc 42	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → (𝑃‘0) < (𝑃‘𝑗)))
36	35	expd 421	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑀) → (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗))))
37	36	impcom 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗)))
38		breq1 5108	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39		fveq2 6893	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
40	39	breq1d 5115	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘0) < (𝑃‘𝑗)))
41	38, 40	imbi12d 347	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → ((𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)) ↔ (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗))))
42	37, 41	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
43	42	expd 421	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑗 ∈ (0..^𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
44	43	com12 33	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
45	1, 30	iccpartlt 48645	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))
46		fveq2 6893	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (𝑃‘𝑗) = (𝑃‘𝑀))
47	39, 46	breqan12rd 5122	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 = 𝑀 ∧ 𝑖 = 0) → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘0) < (𝑃‘𝑀)))
48	45, 47	imbitrrid 249	. . . . . . . . . . . . . 14 ⊢ ((𝑗 = 𝑀 ∧ 𝑖 = 0) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑗)))
49	48	a1dd 51	. . . . . . . . . . . . 13 ⊢ ((𝑗 = 𝑀 ∧ 𝑖 = 0) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
50	49	ex 418	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑀 → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
51	44, 50	jaoi 871	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
52	51	com12 33	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
53		elfzelz 13666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
54	53	ad3antlr 744	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
55	53	peano2zd 12816	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
56	55	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57		elfzoelz 13803	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
58	57	ad2antrr 739	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
59		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
60	57, 53	anim12ci 626	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
61	60	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
62		zltp1le 12756	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
63	61, 62	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
64	59, 63	mpbid 235	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
65	56, 58, 64	3jca 1146	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
66	65	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
67		eluz2 12981	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
68	66, 67	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
69	1	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
70	30	ad2antlr 740	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71		1zzd 12737	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72		elfzelz 13666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℤ)
73	72	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℤ)
74		elfzle1 13670	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75		elfzle1 13670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ≤ 𝑘)
76		1red 11319	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77		elfzel1 13665	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
78	77	zred 12813	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
79	72	zred 12813	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80		letr 11414	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
81	76, 78, 79, 80	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
82	75, 81	mpan2d 707	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (𝑖...𝑗) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
83	74, 82	syl5com 32	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1...𝑀) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
84	83	ad3antlr 744	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
85	84	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ≤ 𝑘)
86		eluz2 12981	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (ℤ≥‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
87	71, 73, 85, 86	syl3anbrc 1362	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ≥‘1))
88		elfzel2 13664	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
89	88	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
90	89	ad2antrr 739	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℤ)
91	79	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℝ)
92	57	zred 12813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
93	92	ad4antr 745	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
94	69	nnred 12360	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95		elfzle2 13671	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ≤ 𝑗)
96	95	adantl 487	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ≤ 𝑗)
97		elfzolt2 13813	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
98	97	ad4antr 745	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
99	91, 93, 94, 96, 98	lelttrd 11478	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100		elfzo2 13806	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
101	87, 90, 99, 100	syl3anbrc 1362	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
102	69, 70, 101	iccpartipre 48642	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃‘𝑘) ∈ ℝ)
103	1	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
104	30	ad2antlr 740	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
105	57	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106		fzoval 13804	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℤ → (𝑖..^𝑗) = (𝑖...(𝑗 − 1)))
107	105, 106	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) = (𝑖...(𝑗 − 1)))
108		elfzo0le 13848	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ≤ 𝑀)
109		0le1 11849	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ≤ 1
110		0red 11321	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111		1red 11319	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
112	53	zred 12813	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113		letr 11414	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114	110, 111, 112, 113	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115	109, 114	mpani 709	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (1...𝑀) → (1 ≤ 𝑖 → 0 ≤ 𝑖))
116	74, 115	mpd 16	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
117	108, 116	anim12ci 626	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀))
118	117	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀))
119		0zd 12715	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120		elfzoel2 13802	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121	119, 120	jca 521	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122	121	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123		ssfzo12bi 13906	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) ∧ (0 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀)))
124	61, 122, 59, 123	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀)))
125	118, 124	mpbird 260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖..^𝑗) ⊆ (0..^𝑀))
126	125	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) ⊆ (0..^𝑀))
127	107, 126	eqsstrrd 3947	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128	127	sselda 3912	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129		iccpartimp 48638	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))))
130	103, 104, 128, 129	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃 ∈ (ℝ* ↑m (0...𝑀)) ∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))))
131	130	simprd 501	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → (𝑃‘𝑘) < (𝑃‘(𝑘 + 1)))
132	54, 68, 102, 131	smonoord 48586	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑗))
133	132	exp31 425	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑗))))
134	133	com23 87	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
135	134	ex 418	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
136		elfzuz 13662	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (ℤ≥‘1))
137	136	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (ℤ≥‘1))
138	88	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
139		simpr 490	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
140		elfzo2 13806	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141	137, 138, 139, 140	syl3anbrc 1362	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
142	1, 30	iccpartiltu 48643	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀))
143		fveq2 6893	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
144	143	breq1d 5115	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀)))
145	144	rspcv 3575	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀)))
146	141, 142, 145	syl2imc 42	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀)))
147	146	expd 421	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑖 < 𝑀 → (𝑃‘𝑖) < (𝑃‘𝑀))))
148	147	impcom 413	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) → (𝑖 < 𝑀 → (𝑃‘𝑖) < (𝑃‘𝑀)))
149	148	imp 412	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀))
150	149	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀)))
151		breq2 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → (𝑖 < 𝑗 ↔ 𝑖 < 𝑀))
152	151	anbi2d 642	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) ↔ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀)))
153	46	breq2d 5117	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘𝑖) < (𝑃‘𝑀)))
154	150, 152, 153	3imtr4d 297	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃‘𝑖) < (𝑃‘𝑗)))
155	154	exp4c 438	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑀 → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
156	135, 155	jaoi 871	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
157	156	com12 33	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
158	52, 157	jaoi 871	. . . . . . . . 9 ⊢ ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
159	158	com13 89	. . . . . . . 8 ⊢ (𝜑 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
160	24, 159	sylbid 243	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
161	160	com3r 88	. . . . . 6 ⊢ ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
162	16, 161	sylbi 220	. . . . 5 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
163	162	com12 33	. . . 4 ⊢ (𝜑 → (𝑖 ∈ ({0} ∪ (1...𝑀)) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
164	12, 163	sylbid 243	. . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
165	164	imp32 424	. 2 ⊢ ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))
166	165	ralrimivva 3208	1 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2146   ≠ wne 2958  ∀wral 3079   ⊆ wss 3899   ∪ cun 4151  {csn 4587   class class class wbr 5105  ‘cfv 6546  (class class class)co 7427   ↑_m cmap 8858  ℝcr 11209  0cc0 11210  1c1 11211   + caddc 11213  ℝ^*cxr 11352   < clt 11353   ≤ cle 11354   − cmin 11551  ℕcn 12345  ℕ₀cn0 12616  ℤcz 12703  ℤ_≥cuz 12975  ...cfz 13649  ..^cfzo 13798  RePartciccp 48634

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-nul 5262  ax-pow 5330  ax-pr 5394  ax-un 7758  ax-cnex 11266  ax-resscn 11267  ax-1cn 11268  ax-icn 11269  ax-addcl 11270  ax-addrcl 11271  ax-mulcl 11272  ax-mulrcl 11273  ax-mulcom 11274  ax-addass 11275  ax-mulass 11276  ax-distr 11277  ax-i2m1 11278  ax-1ne0 11279  ax-1rid 11280  ax-rnegex 11281  ax-rrecex 11282  ax-cnre 11283  ax-pre-lttri 11284  ax-pre-lttrn 11285  ax-pre-ltadd 11286  ax-pre-mulgt0 11287

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7384  df-ov 7430  df-oprab 7431  df-mpo 7432  df-om 7885  df-1st 8008  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-er 8728  df-map 8860  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11355  df-mnf 11356  df-xr 11357  df-ltxr 11358  df-le 11359  df-sub 11553  df-neg 11554  df-nn 12346  df-2 12415  df-n0 12617  df-z 12704  df-uz 12976  df-fz 13650  df-fzo 13799  df-iccp 48635

This theorem is used by:  icceuelpartlem  48656  iccpartnel  48659