Theorem iccpartgt 47537

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 12442	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
3		elnn0uz 12777	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ0 ↔ 𝑀 ∈ (ℤ≥‘0))
4	2, 3	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘0))
5		fzpred 13472	. . . . . . 7 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
6	4, 5	syl 17	. . . . . 6 ⊢ (𝜑 → (0...𝑀) = ({0} ∪ ((0 + 1)...𝑀)))
7		0p1e1 12242	. . . . . . . . 9 ⊢ (0 + 1) = 1
8	7	oveq1i 7356	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
10	9	uneq2d 4115	. . . . . 6 ⊢ (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
11	6, 10	eqtrd 2766	. . . . 5 ⊢ (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
12	11	eleq2d 2817	. . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13		elun 4100	. . . . . . 7 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14		velsn 4589	. . . . . . . 8 ⊢ (𝑖 ∈ {0} ↔ 𝑖 = 0)
15	14	orbi1i 913	. . . . . . 7 ⊢ ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
16	13, 15	bitri 275	. . . . . 6 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17		fzisfzounsn 13680	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
18	4, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
19	18	eleq2d 2817	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20		elun 4100	. . . . . . . . . 10 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21		velsn 4589	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
22	21	orbi2i 912	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
23	20, 22	bitri 275	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
24	19, 23	bitrdi 287	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀)))
25		simpl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (0..^𝑀))
26		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 0 < 𝑗)
27	26	gt0ne0d 11681	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28		fzo1fzo0n0 13615	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 ∈ (1..^𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∧ 𝑗 ≠ 0))
29	25, 27, 28	sylanbrc 583	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (1..^𝑀))
30		iccpartgtprec.p	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
31	1, 30	iccpartigtl 47533	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘))
32		fveq2 6822	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → (𝑃‘𝑘) = (𝑃‘𝑗))
33	32	breq2d 5101	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃‘𝑘) ↔ (𝑃‘0) < (𝑃‘𝑗)))
34	33	rspcv 3568	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘) → (𝑃‘0) < (𝑃‘𝑗)))
35	29, 31, 34	syl2imc 41	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → (𝑃‘0) < (𝑃‘𝑗)))
36	35	expd 415	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑀) → (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗))))
37	36	impcom 407	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗)))
38		breq1 5092	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
40	39	breq1d 5099	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘0) < (𝑃‘𝑗)))
41	38, 40	imbi12d 344	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → ((𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)) ↔ (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗))))
42	37, 41	imbitrrid 246	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
43	42	expd 415	. . . . . . . . . . . . 13 ⊢ (𝑖 = 0 → (𝑗 ∈ (0..^𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
44	43	com12 32	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
45	1, 30	iccpartlt 47534	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))
46		fveq2 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (𝑃‘𝑗) = (𝑃‘𝑀))
47	39, 46	breqan12rd 5106	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 = 𝑀 ∧ 𝑖 = 0) → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘0) < (𝑃‘𝑀)))
48	45, 47	imbitrrid 246	. . . . . . . . . . . . . 14 ⊢ ((𝑗 = 𝑀 ∧ 𝑖 = 0) → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑗)))
49	48	a1dd 50	. . . . . . . . . . . . 13 ⊢ ((𝑗 = 𝑀 ∧ 𝑖 = 0) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
50	49	ex 412	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑀 → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
51	44, 50	jaoi 857	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
52	51	com12 32	. . . . . . . . . 10 ⊢ (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
53		elfzelz 13424	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
54	53	ad3antlr 731	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
55	53	peano2zd 12580	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
56	55	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57		elfzoelz 13559	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
58	57	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
59		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑖 < 𝑗)
60	57, 53	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
61	60	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
62		zltp1le 12522	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
63	61, 62	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 < 𝑗 ↔ (𝑖 + 1) ≤ 𝑗))
64	59, 63	mpbid 232	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ≤ 𝑗)
65	56, 58, 64	3jca 1128	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
66	65	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
67		eluz2 12738	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ≥‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
68	66, 67	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ≥‘(𝑖 + 1)))
69	1	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
70	30	ad2antlr 727	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71		1zzd 12503	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72		elfzelz 13424	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℤ)
73	72	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℤ)
74		elfzle1 13427	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75		elfzle1 13427	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ≤ 𝑘)
76		1red 11113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77		elfzel1 13423	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
78	77	zred 12577	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
79	72	zred 12577	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80		letr 11207	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
81	76, 78, 79, 80	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
82	75, 81	mpan2d 694	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ (𝑖...𝑗) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
83	74, 82	syl5com 31	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1...𝑀) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
84	83	ad3antlr 731	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
85	84	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ≤ 𝑘)
86		eluz2 12738	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (ℤ≥‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
87	71, 73, 85, 86	syl3anbrc 1344	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ≥‘1))
88		elfzel2 13422	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
89	88	ad2antlr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
90	89	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℤ)
91	79	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℝ)
92	57	zred 12577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
93	92	ad4antr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
94	69	nnred 12140	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95		elfzle2 13428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ≤ 𝑗)
96	95	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ≤ 𝑗)
97		elfzolt2 13568	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
98	97	ad4antr 732	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
99	91, 93, 94, 96, 98	lelttrd 11271	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100		elfzo2 13562	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
101	87, 90, 99, 100	syl3anbrc 1344	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
102	69, 70, 101	iccpartipre 47531	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃‘𝑘) ∈ ℝ)
103	1	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
104	30	ad2antlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
105	57	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106		fzoval 13560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑗 ∈ ℤ → (𝑖..^𝑗) = (𝑖...(𝑗 − 1)))
107	105, 106	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) = (𝑖...(𝑗 − 1)))
108		elfzo0le 13603	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ≤ 𝑀)
109		0le1 11640	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ≤ 1
110		0red 11115	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111		1red 11113	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
112	53	zred 12577	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113		letr 11207	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114	110, 111, 112, 113	syl3anc 1373	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115	109, 114	mpani 696	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (1...𝑀) → (1 ≤ 𝑖 → 0 ≤ 𝑖))
116	74, 115	mpd 15	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (1...𝑀) → 0 ≤ 𝑖)
117	108, 116	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀))
118	117	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀))
119		0zd 12480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120		elfzoel2 13558	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121	119, 120	jca 511	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122	121	ad2antrr 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123		ssfzo12bi 13661	. . . . . . . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀)))
125	118, 124	mpbird 257	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖..^𝑗) ⊆ (0..^𝑀))
126	125	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) ⊆ (0..^𝑀))
127	107, 126	eqsstrrd 3965	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128	127	sselda 3929	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129		iccpartimp 47527	. . . . . . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 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 47481	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑗))
133	132	exp31 419	. . . . . . . . . . . . . 14 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝜑 → (𝑃‘𝑖) < (𝑃‘𝑗))))
134	133	com23 86	. . . . . . . . . . . . 13 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
135	134	ex 412	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
136		elfzuz 13420	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (ℤ≥‘1))
137	136	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (ℤ≥‘1))
138	88	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
139		simpr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
140		elfzo2 13562	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141	137, 138, 139, 140	syl3anbrc 1344	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
142	1, 30	iccpartiltu 47532	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀))
143		fveq2 6822	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
144	143	breq1d 5099	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀)))
145	144	rspcv 3568	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1..^𝑀) → (∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀)))
146	141, 142, 145	syl2imc 41	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → (𝑃‘𝑖) < (𝑃‘𝑀)))
147	146	expd 415	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑖 < 𝑀 → (𝑃‘𝑖) < (𝑃‘𝑀))))
148	147	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) → (𝑖 < 𝑀 → (𝑃‘𝑖) < (𝑃‘𝑀)))
149	148	imp 406	. . . . . . . . . . . . . . 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 5093	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑀 → (𝑖 < 𝑗 ↔ 𝑖 < 𝑀))
152	151	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) ↔ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑀)))
153	46	breq2d 5101	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑀 → ((𝑃‘𝑖) < (𝑃‘𝑗) ↔ (𝑃‘𝑖) < (𝑃‘𝑀)))
154	150, 152, 153	3imtr4d 294	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃‘𝑖) < (𝑃‘𝑗)))
155	154	exp4c 432	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑀 → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
156	135, 155	jaoi 857	. . . . . . . . . . 11 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
157	156	com12 32	. . . . . . . . . 10 ⊢ (𝑖 ∈ (1...𝑀) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
158	52, 157	jaoi 857	. . . . . . . . 9 ⊢ ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
159	158	com13 88	. . . . . . . 8 ⊢ (𝜑 → ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
160	24, 159	sylbid 240	. . . . . . 7 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) → ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
161	160	com3r 87	. . . . . 6 ⊢ ((𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
162	16, 161	sylbi 217	. . . . 5 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) → (𝜑 → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
163	162	com12 32	. . . 4 ⊢ (𝜑 → (𝑖 ∈ ({0} ∪ (1...𝑀)) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
164	12, 163	sylbid 240	. . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
165	164	imp32 418	. 2 ⊢ ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))
166	165	ralrimivva 3175	1 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047   ∪ cun 3895   ⊆ wss 3897  {csn 4573   class class class wbr 5089  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750  ℝcr 11005  0cc0 11006  1c1 11007   + caddc 11009  ℝ^*cxr 11145   < clt 11146   ≤ cle 11147   − cmin 11344  ℕcn 12125  ℕ₀cn0 12381  ℤcz 12468  ℤ_≥cuz 12732  ...cfz 13407  ..^cfzo 13554  RePartciccp 47523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-fzo 13555  df-iccp 47524

This theorem is referenced by:  icceuelpartlem  47545  iccpartnel  47548