iccpartgt - Mathbox for Alexander van der Vekens

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Theorem iccpartgt 47408

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 12567	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
3		elnn0uz 12902	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (ℤ_≥‘0))
4	2, 3	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘0))
5		fzpred 13594	. . . . . . 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 12367	. . . . . . . . 9 ⊢ (0 + 1) = 1
8	7	oveq1i 7420	. . . . . . . 8 ⊢ ((0 + 1)...𝑀) = (1...𝑀)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → ((0 + 1)...𝑀) = (1...𝑀))
10	9	uneq2d 4148	. . . . . 6 ⊢ (𝜑 → ({0} ∪ ((0 + 1)...𝑀)) = ({0} ∪ (1...𝑀)))
11	6, 10	eqtrd 2771	. . . . 5 ⊢ (𝜑 → (0...𝑀) = ({0} ∪ (1...𝑀)))
12	11	eleq2d 2821	. . . 4 ⊢ (𝜑 → (𝑖 ∈ (0...𝑀) ↔ 𝑖 ∈ ({0} ∪ (1...𝑀))))
13		elun 4133	. . . . . . 7 ⊢ (𝑖 ∈ ({0} ∪ (1...𝑀)) ↔ (𝑖 ∈ {0} ∨ 𝑖 ∈ (1...𝑀)))
14		velsn 4622	. . . . . . . 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 13800	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (ℤ_≥‘0) → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
18	4, 17	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0...𝑀) = ((0..^𝑀) ∪ {𝑀}))
19	18	eleq2d 2821	. . . . . . . . 9 ⊢ (𝜑 → (𝑗 ∈ (0...𝑀) ↔ 𝑗 ∈ ((0..^𝑀) ∪ {𝑀})))
20		elun 4133	. . . . . . . . . 10 ⊢ (𝑗 ∈ ((0..^𝑀) ∪ {𝑀}) ↔ (𝑗 ∈ (0..^𝑀) ∨ 𝑗 ∈ {𝑀}))
21		velsn 4622	. . . . . . . . . . 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 11806	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑗 ∈ (0..^𝑀) ∧ 0 < 𝑗) → 𝑗 ≠ 0)
28		fzo1fzo0n0 13736	. . . . . . . . . . . . . . . . . . 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 47404	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘0) < (𝑃‘𝑘))
32		fveq2 6881	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑗 → (𝑃‘𝑘) = (𝑃‘𝑗))
33	32	breq2d 5136	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑗 → ((𝑃‘0) < (𝑃‘𝑘) ↔ (𝑃‘0) < (𝑃‘𝑗)))
34	33	rspcv 3602	. . . . . . . . . . . . . . . . . 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 5127	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → (𝑖 < 𝑗 ↔ 0 < 𝑗))
39		fveq2 6881	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 = 0 → (𝑃‘𝑖) = (𝑃‘0))
40	39	breq1d 5134	. . . . . . . . . . . . . . . 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 47405	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃‘0) < (𝑃‘𝑀))
46		fveq2 6881	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑀 → (𝑃‘𝑗) = (𝑃‘𝑀))
47	39, 46	breqan12rd 5141	. . . . . . . . . . . . . . 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 13546	. . . . . . . . . . . . . . . . 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 12705	. . . . . . . . . . . . . . . . . . . 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 13681	. . . . . . . . . . . . . . . . . . . 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 12647	. . . . . . . . . . . . . . . . . . . . 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 12863	. . . . . . . . . . . . . . . . 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 12628	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 1 ∈ ℤ)
72		elfzelz 13546	. . . . . . . . . . . . . . . . . . . 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 13549	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1...𝑀) → 1 ≤ 𝑖)
75		elfzle1 13549	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ≤ 𝑘)
76		1red 11241	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 1 ∈ ℝ)
77		elfzel1 13545	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℤ)
78	77	zred 12702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑖 ∈ ℝ)
79	72	zred 12702	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑗) → 𝑘 ∈ ℝ)
80		letr 11334	. . . . . . . . . . . . . . . . . . . . . . . 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 12863	. . . . . . . . . . . . . . . . . . 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 13544	. . . . . . . . . . . . . . . . . . . 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 12702	. . . . . . . . . . . . . . . . . . . 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 12260	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑀 ∈ ℝ)
95		elfzle2 13550	. . . . . . . . . . . . . . . . . . . 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 13690	. . . . . . . . . . . . . . . . . . . 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 11398	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...𝑗)) → 𝑘 < 𝑀)
100		elfzo2 13684	. . . . . . . . . . . . . . . . . 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 47402	. . . . . . . . . . . . . . . 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 13682	. . . . . . . . . . . . . . . . . . . . 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 13725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑗 ∈ (0..^𝑀) → 𝑗 ≤ 𝑀)
109		0le1 11765	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 0 ≤ 1
110		0red 11243	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 0 ∈ ℝ)
111		1red 11241	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 1 ∈ ℝ)
112	53	zred 12702	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (1...𝑀) → 𝑖 ∈ ℝ)
113		letr 11334	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 12605	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑗 ∈ (0..^𝑀) → 0 ∈ ℤ)
120		elfzoel2 13680	. . . . . . . . . . . . . . . . . . . . . . . . 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 13782	. . . . . . . . . . . . . . . . . . . . . . 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 3999	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) → (𝑖...(𝑗 − 1)) ⊆ (0..^𝑀))
128	127	sselda 3963	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑗 ∈ (0..^𝑀) ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑖 < 𝑗) ∧ 𝜑) ∧ 𝑘 ∈ (𝑖...(𝑗 − 1))) → 𝑘 ∈ (0..^𝑀))
129		iccpartimp 47398	. . . . . . . . . . . . . . . . . 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 47352	. . . . . . . . . . . . . . 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 13542	. . . . . . . . . . . . . . . . . . . . 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 13684	. . . . . . . . . . . . . . . . . . . 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 47403	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ∀𝑘 ∈ (1..^𝑀)(𝑃‘𝑘) < (𝑃‘𝑀))
143		fveq2 6881	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑖 → (𝑃‘𝑘) = (𝑃‘𝑖))
144	143	breq1d 5134	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑖 → ((𝑃‘𝑘) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘𝑀)))
145	144	rspcv 3602	. . . . . . . . . . . . . . . . . . 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 5128	. . . . . . . . . . . . . . 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 5136	. . . . . . . . . . . . . 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 3188	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2933 ∀wral 3052 ∪ cun 3929 ⊆ wss 3931 {csn 4606 class class class wbr 5124 ‘cfv 6536 (class class class)co 7410 ↑_m cmap 8845 ℝcr 11133 0cc0 11134 1c1 11135 + caddc 11137 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 − cmin 11471 ℕcn 12245 ℕ₀cn0 12506 ℤcz 12593 ℤ_≥cuz 12857 ...cfz 13529 ..^cfzo 13676 RePartciccp 47394

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210 ax-pre-mulgt0 11211

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-tr 5235 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6295 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8724 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-nn 12246 df-2 12308 df-n0 12507 df-z 12594 df-uz 12858 df-fz 13530 df-fzo 13677 df-iccp 47395

This theorem is referenced by: icceuelpartlem 47416 iccpartnel 47419

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