iccpartiltu - Mathbox for Alexander van der Vekens

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Theorem iccpartiltu 46675

Description: If there is a partition, then all intermediate points are strictly less than the upper bound. (Contributed by AV, 12-Jul-2020.)

**Hypotheses**
Ref	Expression
iccpartgtprec.m	⊢ (𝜑 → 𝑀 ∈ ℕ)
iccpartgtprec.p	⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))

**Assertion**
Ref	Expression
iccpartiltu	⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))

Distinct variable groups: 𝑖,𝑀 𝑃,𝑖 𝜑,𝑖

Proof of Theorem iccpartiltu Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccpartgtprec.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℕ)
2		ral0 4508	. . . . 5 ⊢ ∀𝑖 ∈ ∅ (𝑃‘𝑖) < (𝑃‘1)
3		oveq2 7422	. . . . . . 7 ⊢ (𝑀 = 1 → (1..^𝑀) = (1..^1))
4		fzo0 13674	. . . . . . 7 ⊢ (1..^1) = ∅
5	3, 4	eqtrdi 2783	. . . . . 6 ⊢ (𝑀 = 1 → (1..^𝑀) = ∅)
6		fveq2 6891	. . . . . . 7 ⊢ (𝑀 = 1 → (𝑃‘𝑀) = (𝑃‘1))
7	6	breq2d 5154	. . . . . 6 ⊢ (𝑀 = 1 → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < (𝑃‘1)))
8	5, 7	raleqbidv 3337	. . . . 5 ⊢ (𝑀 = 1 → (∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀) ↔ ∀𝑖 ∈ ∅ (𝑃‘𝑖) < (𝑃‘1)))
9	2, 8	mpbiri 258	. . . 4 ⊢ (𝑀 = 1 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
10	9	2a1d 26	. . 3 ⊢ (𝑀 = 1 → (𝜑 → (𝑀 ∈ ℕ → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))))
11		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℕ)
12		iccpartgtprec.p	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ (RePart‘𝑀))
13	12	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑃 ∈ (RePart‘𝑀))
14	13	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → 𝑃 ∈ (RePart‘𝑀))
15		nnnn0 12495	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℕ₀)
16		nn0fz0 13617	. . . . . . . . 9 ⊢ (𝑀 ∈ ℕ₀ ↔ 𝑀 ∈ (0...𝑀))
17	15, 16	sylib 217	. . . . . . . 8 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ (0...𝑀))
18	17	adantl 481	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ (0...𝑀))
19	11, 14, 18	iccpartxr 46672	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → (𝑃‘𝑀) ∈ ℝ^*)
20		elxr 13114	. . . . . . 7 ⊢ ((𝑃‘𝑀) ∈ ℝ^* ↔ ((𝑃‘𝑀) ∈ ℝ ∨ (𝑃‘𝑀) = +∞ ∨ (𝑃‘𝑀) = -∞))
21		elfzoelz 13650	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ ℤ)
22	21	ad2antll 728	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑖 ∈ ℤ)
23		elfzo2 13653	. . . . . . . . . . . . . . 15 ⊢ (𝑖 ∈ (1..^𝑀) ↔ (𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀))
24		eluzelz 12848	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ (ℤ_≥‘1) → 𝑖 ∈ ℤ)
25	24	peano2zd 12685	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑖 + 1) ∈ ℤ)
26	25	3ad2ant1 1131	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (𝑖 + 1) ∈ ℤ)
27		simp2 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑀 ∈ ℤ)
28		zltp1le 12628	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑖 < 𝑀 ↔ (𝑖 + 1) ≤ 𝑀))
29	24, 28	sylan 579	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ) → (𝑖 < 𝑀 ↔ (𝑖 + 1) ≤ 𝑀))
30	29	biimp3a 1466	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (𝑖 + 1) ≤ 𝑀)
31		eluz2 12844	. . . . . . . . . . . . . . . 16 ⊢ (𝑀 ∈ (ℤ_≥‘(𝑖 + 1)) ↔ ((𝑖 + 1) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ (𝑖 + 1) ≤ 𝑀))
32	26, 27, 30, 31	syl3anbrc 1341	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → 𝑀 ∈ (ℤ_≥‘(𝑖 + 1)))
33	23, 32	sylbi 216	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑀 ∈ (ℤ_≥‘(𝑖 + 1)))
34	33	ad2antll 728	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑀 ∈ (ℤ_≥‘(𝑖 + 1)))
35		fveq2 6891	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑀 → (𝑃‘𝑘) = (𝑃‘𝑀))
36	35	eqcomd 2733	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑀 → (𝑃‘𝑀) = (𝑃‘𝑘))
37	36	eleq1d 2813	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑀 → ((𝑃‘𝑀) ∈ ℝ ↔ (𝑃‘𝑘) ∈ ℝ))
38	37	biimpcd 248	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑃‘𝑀) ∈ ℝ → (𝑘 = 𝑀 → (𝑃‘𝑘) ∈ ℝ))
39	38	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑘 = 𝑀 → (𝑃‘𝑘) ∈ ℝ))
40	39	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → (𝑘 = 𝑀 → (𝑃‘𝑘) ∈ ℝ))
41	40	com12 32	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑀 → ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → (𝑃‘𝑘) ∈ ℝ))
42	11	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ)
43	42	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑀 ∈ ℕ)
44	43	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → 𝑀 ∈ ℕ)
45	44	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑘 = 𝑀 ∧ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀))) → 𝑀 ∈ ℕ)
46	14	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑃 ∈ (RePart‘𝑀))
47	46	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑃 ∈ (RePart‘𝑀))
48	47	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → 𝑃 ∈ (RePart‘𝑀))
49	48	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑘 = 𝑀 ∧ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀))) → 𝑃 ∈ (RePart‘𝑀))
50		elfz2 13509	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑘 ∈ (𝑖...𝑀) ↔ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)))
51		eluz2 12844	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑖 ∈ (ℤ_≥‘1) ↔ (1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 1 ≤ 𝑖))
52		1red 11231	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → 1 ∈ ℝ)
53		zre 12578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℝ)
54	53	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → 𝑖 ∈ ℝ)
55		zre 12578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑘 ∈ ℤ → 𝑘 ∈ ℝ)
56	55	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℝ)
57		letr 11324	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((1 ∈ ℝ ∧ 𝑖 ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
58	52, 54, 56, 57	syl3anc 1369	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((1 ≤ 𝑖 ∧ 𝑖 ≤ 𝑘) → 1 ≤ 𝑘))
59	58	expcomd 416	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑖 ≤ 𝑘 → (1 ≤ 𝑖 → 1 ≤ 𝑘)))
60	59	adantrd 491	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑖 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀) → (1 ≤ 𝑖 → 1 ≤ 𝑘)))
61	60	3adant2 1129	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀) → (1 ≤ 𝑖 → 1 ≤ 𝑘)))
62	61	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → (1 ≤ 𝑖 → 1 ≤ 𝑘))
63	62	com12 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (1 ≤ 𝑖 → (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → 1 ≤ 𝑘))
64	63	3ad2ant3 1133	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((1 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 1 ≤ 𝑖) → (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → 1 ≤ 𝑘))
65	51, 64	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → 1 ≤ 𝑘))
66	65	3ad2ant1 1131	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ (ℤ_≥‘1) ∧ 𝑀 ∈ ℤ ∧ 𝑖 < 𝑀) → (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → 1 ≤ 𝑘))
67	23, 66	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (1..^𝑀) → (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → 1 ≤ 𝑘))
68	50, 67	biimtrid 241	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑘 ∈ (𝑖...𝑀) → 1 ≤ 𝑘))
69	68	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀)) → 1 ≤ 𝑘)
70	69	3adant3 1130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → 1 ≤ 𝑘)
71		zre 12578	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
72	71, 55	anim12ci 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
73	72	3adant1 1128	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ))
74		ltlen 11331	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑘 ∈ ℝ ∧ 𝑀 ∈ ℝ) → (𝑘 < 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≠ 𝑘)))
75	73, 74	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 < 𝑀 ↔ (𝑘 ≤ 𝑀 ∧ 𝑀 ≠ 𝑘)))
76		nesym 2992	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 ≠ 𝑘 ↔ ¬ 𝑘 = 𝑀)
77	76	anbi2i 622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑘 ≤ 𝑀 ∧ 𝑀 ≠ 𝑘) ↔ (𝑘 ≤ 𝑀 ∧ ¬ 𝑘 = 𝑀))
78	75, 77	bitr2di 288	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 ≤ 𝑀 ∧ ¬ 𝑘 = 𝑀) ↔ 𝑘 < 𝑀))
79	78	biimpd 228	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑘 ≤ 𝑀 ∧ ¬ 𝑘 = 𝑀) → 𝑘 < 𝑀))
80	79	expd 415	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑘 ≤ 𝑀 → (¬ 𝑘 = 𝑀 → 𝑘 < 𝑀)))
81	80	adantld 490	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) → ((𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀) → (¬ 𝑘 = 𝑀 → 𝑘 < 𝑀)))
82	81	imp 406	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑖 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑘 ∈ ℤ) ∧ (𝑖 ≤ 𝑘 ∧ 𝑘 ≤ 𝑀)) → (¬ 𝑘 = 𝑀 → 𝑘 < 𝑀))
83	50, 82	sylbi 216	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑀) → (¬ 𝑘 = 𝑀 → 𝑘 < 𝑀))
84	83	imp 406	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → 𝑘 < 𝑀)
85	84	3adant1 1128	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → 𝑘 < 𝑀)
86	70, 85	jca 511	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → (1 ≤ 𝑘 ∧ 𝑘 < 𝑀))
87		elfzelz 13519	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑀) → 𝑘 ∈ ℤ)
88		1zzd 12609	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑀) → 1 ∈ ℤ)
89		elfzel2 13517	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 ∈ (𝑖...𝑀) → 𝑀 ∈ ℤ)
90	87, 88, 89	3jca 1126	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 ∈ (𝑖...𝑀) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ))
91	90	3ad2ant2 1132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → (𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ))
92		elfzo 13652	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑘 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑘 ∈ (1..^𝑀) ↔ (1 ≤ 𝑘 ∧ 𝑘 < 𝑀)))
93	91, 92	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → (𝑘 ∈ (1..^𝑀) ↔ (1 ≤ 𝑘 ∧ 𝑘 < 𝑀)))
94	86, 93	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...𝑀) ∧ ¬ 𝑘 = 𝑀) → 𝑘 ∈ (1..^𝑀))
95	94	3exp 1117	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑘 ∈ (𝑖...𝑀) → (¬ 𝑘 = 𝑀 → 𝑘 ∈ (1..^𝑀))))
96	95	ad2antll 728	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑘 ∈ (𝑖...𝑀) → (¬ 𝑘 = 𝑀 → 𝑘 ∈ (1..^𝑀))))
97	96	imp 406	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → (¬ 𝑘 = 𝑀 → 𝑘 ∈ (1..^𝑀)))
98	97	impcom 407	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑘 = 𝑀 ∧ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀))) → 𝑘 ∈ (1..^𝑀))
99	45, 49, 98	iccpartipre 46674	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑘 = 𝑀 ∧ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀))) → (𝑃‘𝑘) ∈ ℝ)
100	99	ex 412	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑘 = 𝑀 → ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → (𝑃‘𝑘) ∈ ℝ))
101	41, 100	pm2.61i 182	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...𝑀)) → (𝑃‘𝑘) ∈ ℝ)
102	43	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → 𝑀 ∈ ℕ)
103	47	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → 𝑃 ∈ (RePart‘𝑀))
104		1eluzge0 12892	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ (ℤ_≥‘0)
105		fzoss1 13677	. . . . . . . . . . . . . . . . . . . 20 ⊢ (1 ∈ (ℤ_≥‘0) → (1..^𝑀) ⊆ (0..^𝑀))
106	104, 105	mp1i 13	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → (1..^𝑀) ⊆ (0..^𝑀))
107		elfzoel2 13649	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑀 ∈ ℤ)
108		fzoval 13651	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ∈ ℤ → (𝑖..^𝑀) = (𝑖...(𝑀 − 1)))
109	107, 108	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑖..^𝑀) = (𝑖...(𝑀 − 1)))
110	109	eqcomd 2733	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑖...(𝑀 − 1)) = (𝑖..^𝑀))
111	110	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑘 ∈ (𝑖...(𝑀 − 1)) ↔ 𝑘 ∈ (𝑖..^𝑀)))
112		elfzouz 13654	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (ℤ_≥‘1))
113		fzoss1 13677	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ (ℤ_≥‘1) → (𝑖..^𝑀) ⊆ (1..^𝑀))
114	112, 113	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑖..^𝑀) ⊆ (1..^𝑀))
115	114	sseld 3977	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑘 ∈ (𝑖..^𝑀) → 𝑘 ∈ (1..^𝑀)))
116	111, 115	sylbid 239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑘 ∈ (𝑖...(𝑀 − 1)) → 𝑘 ∈ (1..^𝑀)))
117	116	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → 𝑘 ∈ (1..^𝑀))
118	106, 117	sseldd 3979	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 ∈ (1..^𝑀) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → 𝑘 ∈ (0..^𝑀))
119	118	ex 412	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1..^𝑀) → (𝑘 ∈ (𝑖...(𝑀 − 1)) → 𝑘 ∈ (0..^𝑀)))
120	119	ad2antll 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑘 ∈ (𝑖...(𝑀 − 1)) → 𝑘 ∈ (0..^𝑀)))
121	120	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → 𝑘 ∈ (0..^𝑀))
122		iccpartimp 46670	. . . . . . . . . . . . . . 15 ⊢ ((𝑀 ∈ ℕ ∧ 𝑃 ∈ (RePart‘𝑀) ∧ 𝑘 ∈ (0..^𝑀)) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))))
123	102, 103, 121, 122	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → (𝑃 ∈ (ℝ^* ↑_m (0...𝑀)) ∧ (𝑃‘𝑘) < (𝑃‘(𝑘 + 1))))
124	123	simprd 495	. . . . . . . . . . . . 13 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) ∧ 𝑘 ∈ (𝑖...(𝑀 − 1))) → (𝑃‘𝑘) < (𝑃‘(𝑘 + 1)))
125	22, 34, 101, 124	smonoord 46624	. . . . . . . . . . . 12 ⊢ (((𝑃‘𝑀) ∈ ℝ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑃‘𝑖) < (𝑃‘𝑀))
126	125	ex 412	. . . . . . . . . . 11 ⊢ ((𝑃‘𝑀) ∈ ℝ → ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)))
127		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑖 ∈ (1..^𝑀))
128	42, 46, 127	iccpartipre 46674	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) ∈ ℝ)
129		ltpnf 13118	. . . . . . . . . . . . 13 ⊢ ((𝑃‘𝑖) ∈ ℝ → (𝑃‘𝑖) < +∞)
130	128, 129	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < +∞)
131		breq2 5146	. . . . . . . . . . . 12 ⊢ ((𝑃‘𝑀) = +∞ → ((𝑃‘𝑖) < (𝑃‘𝑀) ↔ (𝑃‘𝑖) < +∞))
132	130, 131	imbitrrid 245	. . . . . . . . . . 11 ⊢ ((𝑃‘𝑀) = +∞ → ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)))
133	42	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑀 ∈ ℕ)
134	46	adantl 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑃 ∈ (RePart‘𝑀))
135		elfzofz 13666	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ (1..^𝑀) → 𝑖 ∈ (1...𝑀))
136	135	ad2antll 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑖 ∈ (1...𝑀))
137		elfzubelfz 13531	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ (1...𝑀) → 𝑀 ∈ (1...𝑀))
138	136, 137	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → 𝑀 ∈ (1...𝑀))
139	133, 134, 138	iccpartgtprec 46673	. . . . . . . . . . . . . 14 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑃‘(𝑀 − 1)) < (𝑃‘𝑀))
140		breq2 5146	. . . . . . . . . . . . . . . 16 ⊢ (-∞ = (𝑃‘𝑀) → ((𝑃‘(𝑀 − 1)) < -∞ ↔ (𝑃‘(𝑀 − 1)) < (𝑃‘𝑀)))
141	140	eqcoms 2735	. . . . . . . . . . . . . . 15 ⊢ ((𝑃‘𝑀) = -∞ → ((𝑃‘(𝑀 − 1)) < -∞ ↔ (𝑃‘(𝑀 − 1)) < (𝑃‘𝑀)))
142	141	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → ((𝑃‘(𝑀 − 1)) < -∞ ↔ (𝑃‘(𝑀 − 1)) < (𝑃‘𝑀)))
143	139, 142	mpbird 257	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑃‘(𝑀 − 1)) < -∞)
144	15	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → 𝑀 ∈ ℕ₀)
145	144	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → 𝑀 ∈ ℕ₀)
146		nnne0 12262	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑀 ∈ ℕ → 𝑀 ≠ 0)
147	146	adantl 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → 𝑀 ≠ 0)
148		df-ne 2936	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑀 ≠ 1 ↔ ¬ 𝑀 = 1)
149	148	biimpri 227	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (¬ 𝑀 = 1 → 𝑀 ≠ 1)
150	149	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → 𝑀 ≠ 1)
151	150	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → 𝑀 ≠ 1)
152	144, 147, 151	3jca 1126	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0 ∧ 𝑀 ≠ 1))
153	152	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0 ∧ 𝑀 ≠ 1))
154		nn0n0n1ge2 12555	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ ℕ₀ ∧ 𝑀 ≠ 0 ∧ 𝑀 ≠ 1) → 2 ≤ 𝑀)
155	153, 154	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → 2 ≤ 𝑀)
156	145, 155	jca 511	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑀 ∈ ℕ₀ ∧ 2 ≤ 𝑀))
157	156	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑀 ∈ ℕ₀ ∧ 2 ≤ 𝑀))
158		ige2m1fz 13609	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℕ₀ ∧ 2 ≤ 𝑀) → (𝑀 − 1) ∈ (0...𝑀))
159	157, 158	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑀 − 1) ∈ (0...𝑀))
160	133, 134, 159	iccpartxr 46672	. . . . . . . . . . . . . 14 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑃‘(𝑀 − 1)) ∈ ℝ^*)
161		nltmnf 13127	. . . . . . . . . . . . . 14 ⊢ ((𝑃‘(𝑀 − 1)) ∈ ℝ^* → ¬ (𝑃‘(𝑀 − 1)) < -∞)
162	160, 161	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → ¬ (𝑃‘(𝑀 − 1)) < -∞)
163	143, 162	pm2.21dd 194	. . . . . . . . . . . 12 ⊢ (((𝑃‘𝑀) = -∞ ∧ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀))) → (𝑃‘𝑖) < (𝑃‘𝑀))
164	163	ex 412	. . . . . . . . . . 11 ⊢ ((𝑃‘𝑀) = -∞ → ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)))
165	126, 132, 164	3jaoi 1425	. . . . . . . . . 10 ⊢ (((𝑃‘𝑀) ∈ ℝ ∨ (𝑃‘𝑀) = +∞ ∨ (𝑃‘𝑀) = -∞) → ((((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀)))
166	165	impl 455	. . . . . . . . 9 ⊢ (((((𝑃‘𝑀) ∈ ℝ ∨ (𝑃‘𝑀) = +∞ ∨ (𝑃‘𝑀) = -∞) ∧ ((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ)) ∧ 𝑖 ∈ (1..^𝑀)) → (𝑃‘𝑖) < (𝑃‘𝑀))
167	166	ralrimiva 3141	. . . . . . . 8 ⊢ ((((𝑃‘𝑀) ∈ ℝ ∨ (𝑃‘𝑀) = +∞ ∨ (𝑃‘𝑀) = -∞) ∧ ((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ)) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
168	167	ex 412	. . . . . . 7 ⊢ (((𝑃‘𝑀) ∈ ℝ ∨ (𝑃‘𝑀) = +∞ ∨ (𝑃‘𝑀) = -∞) → (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)))
169	20, 168	sylbi 216	. . . . . 6 ⊢ ((𝑃‘𝑀) ∈ ℝ^* → (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)))
170	19, 169	mpcom 38	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑀 = 1) ∧ 𝑀 ∈ ℕ) → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))
171	170	ex 412	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 = 1) → (𝑀 ∈ ℕ → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)))
172	171	expcom 413	. . 3 ⊢ (¬ 𝑀 = 1 → (𝜑 → (𝑀 ∈ ℕ → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))))
173	10, 172	pm2.61i 182	. 2 ⊢ (𝜑 → (𝑀 ∈ ℕ → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀)))
174	1, 173	mpd 15	1 ⊢ (𝜑 → ∀𝑖 ∈ (1..^𝑀)(𝑃‘𝑖) < (𝑃‘𝑀))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ w3o 1084 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ⊆ wss 3944 ∅c0 4318 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ↑_m cmap 8834 ℝcr 11123 0cc0 11124 1c1 11125 + caddc 11127 +∞cpnf 11261 -∞cmnf 11262 ℝ^*cxr 11263 < clt 11264 ≤ cle 11265 − cmin 11460 ℕcn 12228 2c2 12283 ℕ₀cn0 12488 ℤcz 12574 ℤ_≥cuz 12838 ...cfz 13502 ..^cfzo 13645 RePartciccp 46666

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-map 8836 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-nn 12229 df-2 12291 df-n0 12489 df-z 12575 df-uz 12839 df-fz 13503 df-fzo 13646 df-iccp 46667

This theorem is referenced by: iccpartlt 46677 iccpartltu 46678 iccpartgt 46680

W3C validator