iccpartgt - Mathbox for Alexander van der Vekens

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Theorem iccpartgt 45609

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 12473	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
3		elnn0uz 12808	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (ℤ_≥‘0))
4	2, 3	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
5		fzpred 13489	. . . . . . 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 12275	. . . . . . . . 9 ⊢ (0 + 1) = 1
8	7	oveq1i 7367	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
10	9	uneq2d 4123	. . . . . 6 ⊢ (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
11	6, 10	eqtrd 2776	. . . . 5 ⊢ (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
12	11	eleq2d 2823	. . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13		elun 4108	. . . . . . 7 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14		velsn 4602	. . . . . . . 8 ⊢ (𝑖 ∈ {0} ↔ 𝑖 = 0)
15	14	orbi1i 912	. . . . . . 7 ⊢ ((𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
16	13, 15	bitri 274	. . . . . 6 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 = 0 ∨ 𝑖 ∈ (1...𝑀)))
17		fzisfzounsn 13684	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
18	4, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
19	18	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20		elun 4108	. . . . . . . . . 10 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21		velsn 4602	. . . . . . . . . . 11 ⊢ (𝑗 ∈ {𝑀} ↔ 𝑗 = 𝑀)
22	21	orbi2i 911	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
23	20, 22	bitri 274	. . . . . . . . 9 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀))
24	19, 23	bitrdi 286	. . . . . . . 8 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 = 𝑀)))
25		simpl 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ∈ (0..^𝑀))
26		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 0 < 𝑗)
27	26	gt0ne0d 11719	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28		fzo1fzo0n0 13623	. . . . . . . . . . . . . . . . . . 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 45605	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘))
32		fveq2 6842	. . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑗 ∈ (0..^𝑀) → (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗))))
37	36	impcom 408	. . . . . . . . . . . . . . 15 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗)))
38		breq1 5108	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39		fveq2 6842	. . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . 15 ⊢ (𝑖 = 0 → ((𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)) ↔ (0 < 𝑗 → (𝑃‘0) < (𝑃‘𝑗))))
42	37, 41	syl5ibr 245	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → ((𝑗 ∈ (0..^𝑀) ∧ 𝜑) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗))))
43	42	expd 416	. . . . . . . . . . . . 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 45606	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))
46		fveq2 6842	. . . . . . . . . . . . . . . 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	syl5ibr 245	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑀 → (𝑖 = 0 → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
51	44, 50	jaoi 855	. . . . . . . . . . 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 13441	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℤ)
54	53	ad3antlr 729	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑖 ∈ ℤ)
55	53	peano2zd 12610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑀) → (𝑖 + 1) ∈ ℤ)
56	55	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 + 1) ∈ ℤ)
57		elfzoelz 13572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℤ)
58	57	ad2antrr 724	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑗 ∈ ℤ)
59		simpr 485	. . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ))
62		zltp1le 12553	. . . . . . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
67		eluz2 12769	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 ∈ (ℤ_≥‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑗 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑗))
68	66, 67	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ (ℤ_≥‘(𝑖 + 1)))
69	1	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℕ)
70	30	ad2antlr 725	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑃 ∈ (RePart‘𝑀))
71		1zzd 12534	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72		elfzelz 13441	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℤ)
73	72	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℤ)
74		elfzle1 13444	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75		elfzle1 13444	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ≤ 𝑘)
76		1red 11156	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77		elfzel1 13440	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
78	77	zred 12607	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
79	72	zred 12607	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80		letr 11249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
81	76, 78, 79, 80	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
82	75, 81	mpan2d 692	. . . . . . . . . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑘 ∈ (𝑖...𝑗) → 1 ≤ 𝑘))
85	84	imp 407	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ≤ 𝑘)
86		eluz2 12769	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 𝑘 ∈ ℤ ∧ 1 ≤ 𝑘))
87	71, 73, 85, 86	syl3anbrc 1343	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (ℤ_≥‘1))
88		elfzel2 13439	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ ℤ)
89	88	ad2antlr 725	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → 𝑀 ∈ ℤ)
90	89	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℤ)
91	79	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ ℝ)
92	57	zred 12607	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ∈ ℝ)
93	92	ad4antr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 ∈ ℝ)
94	69	nnred 12168	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95		elfzle2 13445	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ≤ 𝑗)
96	95	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ≤ 𝑗)
97		elfzolt2 13581	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 < 𝑀)
98	97	ad4antr 730	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑗 < 𝑀)
99	91, 93, 94, 96, 98	lelttrd 11313	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100		elfzo2 13575	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ (1..^𝑀) ↔ (𝑘 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑘 < 𝑀))
101	87, 90, 99, 100	syl3anbrc 1343	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 ∈ (1..^𝑀))
102	69, 70, 101	iccpartipre 45603	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → (𝑃‘𝑘) ∈ ℝ)
103	1	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑀 ∈ ℕ)
104	30	ad2antlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑃 ∈ (RePart‘𝑀))
105	57	ad3antrrr 728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → 𝑗 ∈ ℤ)
106		fzoval 13573	. . . . . . . . . . . . . . . . . . . . 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 13616	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ≤ 𝑀)
109		0le1 11678	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ≤ 1
110		0red 11158	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111		1red 11156	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
112	53	zred 12607	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113		letr 11249	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑖 ∈ ℝ) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
114	110, 111, 112, 113	syl3anc 1371	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (1...𝑀) → ((0 ≤ 1 ∧ 1 ≤ 𝑖) → 0 ≤ 𝑖))
115	109, 114	mpani 694	. . . . . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀))
119		0zd 12511	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120		elfzoel2 13571	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑀 ∈ ℤ)
121	119, 120	jca 512	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
122	121	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (0 ∈ ℤ ∧ 𝑀 ∈ ℤ))
123		ssfzo12bi 13667	. . . . . . . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → ((𝑖..^𝑗) ⊆ (0..^𝑀) ↔ (0 ≤ 𝑖 ∧ 𝑗 ≤ 𝑀)))
125	118, 124	mpbird 256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) → (𝑖..^𝑗) ⊆ (0..^𝑀))
126	125	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖..^𝑗) ⊆ (0..^𝑀))
127	107, 126	eqsstrrd 3983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128	127	sselda 3944	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129		iccpartimp 45599	. . . . . . . . . . . . . . . . . 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 1371	. . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 45553	. . . . . . . . . . . . . . 15 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑃‘𝑖) < (𝑃‘𝑗))
133	132	exp31 420	. . . . . . . . . . . . . 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 413	. . . . . . . . . . . 12 ⊢ (𝑗 ∈ (0..^𝑀) → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
136		elfzuz 13437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ (ℤ_≥‘1))
137	136	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (ℤ_≥‘1))
138	88	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
139		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 < 𝑀)
140		elfzo2 13575	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
141	137, 138, 139, 140	syl3anbrc 1343	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝑖 < 𝑀) → 𝑖 ∈ (1..^𝑀))
142	1, 30	iccpartiltu 45604	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀))
143		fveq2 6842	. . . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . . 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 416	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑖 ∈ (1...𝑀) → (𝑖 < 𝑀 → (𝑃‘𝑖) < (𝑃‘𝑀))))
148	147	impcom 408	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (1...𝑀) ∧ 𝜑) → (𝑖 < 𝑀 → (𝑃‘𝑖) < (𝑃‘𝑀)))
149	148	imp 407	. . . . . . . . . . . . . . 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 629	. . . . . . . . . . . . . 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 293	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝑀 → (((𝑖 ∈ (1...𝑀) ∧ 𝜑) ∧ 𝑖 < 𝑗) → (𝑃‘𝑖) < (𝑃‘𝑗)))
155	154	exp4c 433	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝑀 → (𝑖 ∈ (1...𝑀) → (𝜑 → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
156	135, 155	jaoi 855	. . . . . . . . . . 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 855	. . . . . . . . 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 239	. . . . . . 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 216	. . . . 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 239	. . 3 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) → (𝑗 ∈ (0...𝑀) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))))
165	164	imp32 419	. 2 ⊢ ((𝜑 ∧ (𝑖 ∈ (0...𝑀) ∧ 𝑗 ∈ (0...𝑀))) → (𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))
166	165	ralrimivva 3197	1 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝑀)∀𝑗 ∈ (0...𝑀)(𝑖 < 𝑗 → (𝑃‘𝑖) < (𝑃‘𝑗)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∪ cun 3908 ⊆ wss 3910 {csn 4586 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 ℝcr 11050 0cc0 11051 1c1 11052 + caddc 11054 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 ℕcn 12153 ℕ₀cn0 12413 ℤcz 12499 ℤ_≥cuz 12763 ...cfz 13424 ..^cfzo 13567 RePartciccp 45595

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-iccp 45596

This theorem is referenced by: icceuelpartlem 45617 iccpartnel 45620

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