iccelpart - Mathbox for Alexander van der Vekens

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Theorem iccelpart 44842

Description: An element of any partitioned half-open interval of extended reals is an element of a part of this partition. (Contributed by AV, 18-Jul-2020.)

**Assertion**
Ref	Expression
iccelpart	⊢ (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))

Distinct variable groups: 𝑖,𝑀,𝑝 𝑖,𝑋,𝑝

Proof of Theorem iccelpart Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6768	. . 3 ⊢ (𝑥 = 1 → (RePart‘𝑥) = (RePart‘1))
2		fveq2 6768	. . . . . 6 ⊢ (𝑥 = 1 → (𝑝‘𝑥) = (𝑝‘1))
3	2	oveq2d 7285	. . . . 5 ⊢ (𝑥 = 1 → ((𝑝‘0)[,)(𝑝‘𝑥)) = ((𝑝‘0)[,)(𝑝‘1)))
4	3	eleq2d 2824	. . . 4 ⊢ (𝑥 = 1 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
5		oveq2 7277	. . . . . 6 ⊢ (𝑥 = 1 → (0..^𝑥) = (0..^1))
6		fzo01 13458	. . . . . 6 ⊢ (0..^1) = {0}
7	5, 6	eqtrdi 2794	. . . . 5 ⊢ (𝑥 = 1 → (0..^𝑥) = {0})
8	7	rexeqdv 3348	. . . 4 ⊢ (𝑥 = 1 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
9	4, 8	imbi12d 345	. . 3 ⊢ (𝑥 = 1 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
10	1, 9	raleqbidv 3335	. 2 ⊢ (𝑥 = 1 → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
11		fveq2 6768	. . 3 ⊢ (𝑥 = 𝑦 → (RePart‘𝑥) = (RePart‘𝑦))
12		fveq2 6768	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑝‘𝑥) = (𝑝‘𝑦))
13	12	oveq2d 7285	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑝‘0)[,)(𝑝‘𝑥)) = ((𝑝‘0)[,)(𝑝‘𝑦)))
14	13	eleq2d 2824	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))))
15		oveq2 7277	. . . . 5 ⊢ (𝑥 = 𝑦 → (0..^𝑥) = (0..^𝑦))
16	15	rexeqdv 3348	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
17	14, 16	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
18	11, 17	raleqbidv 3335	. 2 ⊢ (𝑥 = 𝑦 → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
19		fveq2 6768	. . 3 ⊢ (𝑥 = (𝑦 + 1) → (RePart‘𝑥) = (RePart‘(𝑦 + 1)))
20		fveq2 6768	. . . . . 6 ⊢ (𝑥 = (𝑦 + 1) → (𝑝‘𝑥) = (𝑝‘(𝑦 + 1)))
21	20	oveq2d 7285	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → ((𝑝‘0)[,)(𝑝‘𝑥)) = ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))
22	21	eleq2d 2824	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1)))))
23		oveq2 7277	. . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (0..^𝑥) = (0..^(𝑦 + 1)))
24	23	rexeqdv 3348	. . . 4 ⊢ (𝑥 = (𝑦 + 1) → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
25	22, 24	imbi12d 345	. . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
26	19, 25	raleqbidv 3335	. 2 ⊢ (𝑥 = (𝑦 + 1) → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘(𝑦 + 1))(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
27		fveq2 6768	. . 3 ⊢ (𝑥 = 𝑀 → (RePart‘𝑥) = (RePart‘𝑀))
28		fveq2 6768	. . . . . 6 ⊢ (𝑥 = 𝑀 → (𝑝‘𝑥) = (𝑝‘𝑀))
29	28	oveq2d 7285	. . . . 5 ⊢ (𝑥 = 𝑀 → ((𝑝‘0)[,)(𝑝‘𝑥)) = ((𝑝‘0)[,)(𝑝‘𝑀)))
30	29	eleq2d 2824	. . . 4 ⊢ (𝑥 = 𝑀 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀))))
31		oveq2 7277	. . . . 5 ⊢ (𝑥 = 𝑀 → (0..^𝑥) = (0..^𝑀))
32	31	rexeqdv 3348	. . . 4 ⊢ (𝑥 = 𝑀 → (∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
33	30, 32	imbi12d 345	. . 3 ⊢ (𝑥 = 𝑀 → ((𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
34	27, 33	raleqbidv 3335	. 2 ⊢ (𝑥 = 𝑀 → (∀𝑝 ∈ (RePart‘𝑥)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑥)) → ∃𝑖 ∈ (0..^𝑥)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
35		0nn0 12237	. . . . 5 ⊢ 0 ∈ ℕ₀
36		fveq2 6768	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑝‘𝑖) = (𝑝‘0))
37		fv0p1e1 12085	. . . . . . . 8 ⊢ (𝑖 = 0 → (𝑝‘(𝑖 + 1)) = (𝑝‘1))
38	36, 37	oveq12d 7287	. . . . . . 7 ⊢ (𝑖 = 0 → ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝‘0)[,)(𝑝‘1)))
39	38	eleq2d 2824	. . . . . 6 ⊢ (𝑖 = 0 → (𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
40	39	rexsng 4612	. . . . 5 ⊢ (0 ∈ ℕ₀ → (∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1))))
41	35, 40	ax-mp 5	. . . 4 ⊢ (∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)))
42	41	biimpri 227	. . 3 ⊢ (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
43	42	rgenw 3076	. 2 ⊢ ∀𝑝 ∈ (RePart‘1)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘1)) → ∃𝑖 ∈ {0}𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
44		nfv 1917	. . . . 5 ⊢ Ⅎ𝑝 𝑦 ∈ ℕ
45		nfra1 3144	. . . . 5 ⊢ Ⅎ𝑝∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
46	44, 45	nfan 1902	. . . 4 ⊢ Ⅎ𝑝(𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
47		nnnn0 12229	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℕ₀)
48		fzonn0p1 13453	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ₀ → 𝑦 ∈ (0..^(𝑦 + 1)))
49	47, 48	syl 17	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (0..^(𝑦 + 1)))
50	49	ad2antrr 723	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑦 ∈ (0..^(𝑦 + 1)))
51		fveq2 6768	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑦 → (𝑝‘𝑖) = (𝑝‘𝑦))
52		fvoveq1 7292	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑦 → (𝑝‘(𝑖 + 1)) = (𝑝‘(𝑦 + 1)))
53	51, 52	oveq12d 7287	. . . . . . . . . 10 ⊢ (𝑖 = 𝑦 → ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = ((𝑝‘𝑦)[,)(𝑝‘(𝑦 + 1))))
54	53	eleq2d 2824	. . . . . . . . 9 ⊢ (𝑖 = 𝑦 → (𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘𝑦)[,)(𝑝‘(𝑦 + 1)))))
55	54	adantl 482	. . . . . . . 8 ⊢ ((((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) ∧ 𝑖 = 𝑦) → (𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘𝑦)[,)(𝑝‘(𝑦 + 1)))))
56		peano2nn 11974	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
57	56	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ ℕ)
58		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑝 ∈ (RePart‘(𝑦 + 1)))
59	56	nnnn0d 12282	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ₀)
60		0elfz 13342	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + 1) ∈ ℕ₀ → 0 ∈ (0...(𝑦 + 1)))
61	59, 60	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → 0 ∈ (0...(𝑦 + 1)))
62	61	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...(𝑦 + 1)))
63	57, 58, 62	iccpartxr 44828	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) ∈ ℝ^*)
64		nn0fz0 13343	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 + 1) ∈ ℕ₀ ↔ (𝑦 + 1) ∈ (0...(𝑦 + 1)))
65	59, 64	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
66	65	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑦 + 1) ∈ (0...(𝑦 + 1)))
67	57, 58, 66	iccpartxr 44828	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘(𝑦 + 1)) ∈ ℝ^*)
68	63, 67	jca 512	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ^* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ^*))
69	68	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ^* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ^*))
70		elico1 13111	. . . . . . . . . . . 12 ⊢ (((𝑝‘0) ∈ ℝ^* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ^) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ^ ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
71	69, 70	syl 17	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
72		simp1 1135	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ℝ^*)
73	72	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑝‘𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ^*)
74		simpl 483	. . . . . . . . . . . . . . 15 ⊢ (((𝑝‘𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝‘𝑦) ≤ 𝑋)
75		simpr3 1195	. . . . . . . . . . . . . . 15 ⊢ (((𝑝‘𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝‘(𝑦 + 1)))
76	73, 74, 75	3jca 1127	. . . . . . . . . . . . . 14 ⊢ (((𝑝‘𝑦) ≤ 𝑋 ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))))
77	76	ex 413	. . . . . . . . . . . . 13 ⊢ ((𝑝‘𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
78	77	adantl 482	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
79	78	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
80	71, 79	sylbid 239	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
81	80	impr 455	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))))
82		nn0fz0 13343	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℕ₀ ↔ 𝑦 ∈ (0...𝑦))
83	47, 82	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (0...𝑦))
84		fzelp1 13297	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (0...𝑦) → 𝑦 ∈ (0...(𝑦 + 1)))
85	83, 84	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (0...(𝑦 + 1)))
86	85	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...(𝑦 + 1)))
87	57, 58, 86	iccpartxr 44828	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘𝑦) ∈ ℝ^*)
88	87, 67	jca 512	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘𝑦) ∈ ℝ^* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ^*))
89	88	ad2ant2r 744	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ((𝑝‘𝑦) ∈ ℝ^* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ^*))
90		elico1 13111	. . . . . . . . . 10 ⊢ (((𝑝‘𝑦) ∈ ℝ^* ∧ (𝑝‘(𝑦 + 1)) ∈ ℝ^) → (𝑋 ∈ ((𝑝‘𝑦)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ^ ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
91	89, 90	syl 17	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → (𝑋 ∈ ((𝑝‘𝑦)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
92	81, 91	mpbird 256	. . . . . . . 8 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → 𝑋 ∈ ((𝑝‘𝑦)[,)(𝑝‘(𝑦 + 1))))
93	50, 55, 92	rspcedvd 3564	. . . . . . 7 ⊢ (((𝑦 ∈ ℕ ∧ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑝 ∈ (RePart‘(𝑦 + 1)) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
94	93	exp43 437	. . . . . 6 ⊢ (𝑦 ∈ ℕ → ((𝑝‘𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
95	94	adantr 481	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → ((𝑝‘𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
96		iccpartres 44827	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦))
97		rspsbca 3814	. . . . . . . . . . . 12 ⊢ (((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → [(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
98		vex 3435	. . . . . . . . . . . . . . 15 ⊢ 𝑝 ∈ V
99	98	resex 5934	. . . . . . . . . . . . . 14 ⊢ (𝑝 ↾ (0...𝑦)) ∈ V
100		sbcimg 3768	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → [(𝑝 ↾ (0...𝑦)) / 𝑝]∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
101		sbcel2 4351	. . . . . . . . . . . . . . . . 17 ⊢ ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) ↔ 𝑋 ∈ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘0)[,)(𝑝‘𝑦)))
102		csbov12g 7313	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘0)[,)(𝑝‘𝑦)) = (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘0)[,)⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑦)))
103		csbfv12 6811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘0) = (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌0)
104		csbvarg 4367	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝 = (𝑝 ↾ (0...𝑦)))
105		csbconstg 3852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌0 = 0)
106	104, 105	fveq12d 6775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌0) = ((𝑝 ↾ (0...𝑦))‘0))
107	103, 106	eqtrid 2790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
108		csbfv12 6811	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑦) = (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑦)
109		csbconstg 3852	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑦 = 𝑦)
110	104, 109	fveq12d 6775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
111	108, 110	eqtrid 2790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
112	107, 111	oveq12d 7287	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘0)[,)⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
113	102, 112	eqtrd 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘0)[,)(𝑝‘𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
114	113	eleq2d 2824	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋 ∈ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘0)[,)(𝑝‘𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
115	101, 114	syl5bb 283	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
116		sbcrex 3809	. . . . . . . . . . . . . . . . 17 ⊢ ([(𝑝 ↾ (0...𝑦)) / 𝑝]∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)[(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
117		sbcel2 4351	. . . . . . . . . . . . . . . . . . 19 ⊢ ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
118		csbov12g 7313	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑖)[,)⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘(𝑖 + 1))))
119		csbfv12 6811	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑖) = (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑖)
120		csbconstg 3852	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑖 = 𝑖)
121	104, 120	fveq12d 6775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
122	119, 121	eqtrid 2790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑖) = ((𝑝 ↾ (0...𝑦))‘𝑖))
123		csbfv12 6811	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘(𝑖 + 1)) = (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑖 + 1))
124		csbconstg 3852	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑖 + 1) = (𝑖 + 1))
125	104, 124	fveq12d 6775	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌𝑝‘⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
126	123, 125	eqtrid 2790	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘(𝑖 + 1)) = ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))
127	122, 126	oveq12d 7287	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘𝑖)[,)⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
128	118, 127	eqtrd 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) = (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))
129	128	eleq2d 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (𝑋 ∈ ⦋(𝑝 ↾ (0...𝑦)) / 𝑝⦌((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
130	117, 129	syl5bb 283	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
131	130	rexbidv 3225	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (∃𝑖 ∈ (0..^𝑦)[(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
132	116, 131	syl5bb 283	. . . . . . . . . . . . . . . 16 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝]∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
133	115, 132	imbi12d 345	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → (([(𝑝 ↾ (0...𝑦)) / 𝑝]𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → [(𝑝 ↾ (0...𝑦)) / 𝑝]∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))))
134	100, 133	bitrd 278	. . . . . . . . . . . . . 14 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ V → ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))))))
135	99, 134	ax-mp 5	. . . . . . . . . . . . 13 ⊢ ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) ↔ (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))))
136	68, 70	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
137	136	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) ↔ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))))
138	72	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ℝ^*)
139		simpr2 1194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑝‘0) ≤ 𝑋)
140		xrltnle 11031	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑋 ∈ ℝ^* ∧ (𝑝‘𝑦) ∈ ℝ^*) → (𝑋 < (𝑝‘𝑦) ↔ ¬ (𝑝‘𝑦) ≤ 𝑋))
141	72, 87, 140	syl2anr 597	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 < (𝑝‘𝑦) ↔ ¬ (𝑝‘𝑦) ≤ 𝑋))
142	141	exbiri 808	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → 𝑋 < (𝑝‘𝑦))))
143	142	com23 86	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 < (𝑝‘𝑦))))
144	143	imp31 418	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 < (𝑝‘𝑦))
145	138, 139, 144	3jca 1127	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘𝑦)))
146	63, 87	jca 512	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0) ∈ ℝ^* ∧ (𝑝‘𝑦) ∈ ℝ^*))
147	146	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → ((𝑝‘0) ∈ ℝ^* ∧ (𝑝‘𝑦) ∈ ℝ^*))
148		elico1 13111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑝‘0) ∈ ℝ^* ∧ (𝑝‘𝑦) ∈ ℝ^) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) ↔ (𝑋 ∈ ℝ^ ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘𝑦))))
149	147, 148	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) ↔ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘𝑦))))
150	145, 149	mpbird 256	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) ∧ (𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1)))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)))
151	150	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) → ((𝑋 ∈ ℝ^* ∧ (𝑝‘0) ≤ 𝑋 ∧ 𝑋 < (𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))))
152	137, 151	sylbid 239	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))))
153		0elfz 13342	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑦 ∈ ℕ₀ → 0 ∈ (0...𝑦))
154	47, 153	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑦 ∈ ℕ → 0 ∈ (0...𝑦))
155	154	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 0 ∈ (0...𝑦))
156		fvres 6787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (0 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘0) = (𝑝‘0))
157	155, 156	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝 ↾ (0...𝑦))‘0) = (𝑝‘0))
158	157	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘0) = ((𝑝 ↾ (0...𝑦))‘0))
159	83	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → 𝑦 ∈ (0...𝑦))
160		fvres 6787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘𝑦) = (𝑝‘𝑦))
161	159, 160	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝 ↾ (0...𝑦))‘𝑦) = (𝑝‘𝑦))
162	161	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑝‘𝑦) = ((𝑝 ↾ (0...𝑦))‘𝑦))
163	158, 162	oveq12d 7287	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → ((𝑝‘0)[,)(𝑝‘𝑦)) = (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
164	163	eleq2d 2824	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) ↔ 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦))))
165	164	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) → 𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)))
166		elfzofz 13392	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑖 ∈ (0..^𝑦) → 𝑖 ∈ (0...𝑦))
167	166	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → 𝑖 ∈ (0...𝑦))
168		fvres 6787	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑖 ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘𝑖) = (𝑝‘𝑖))
169	167, 168	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘𝑖) = (𝑝‘𝑖))
170		fzofzp1 13473	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑖 ∈ (0..^𝑦) → (𝑖 + 1) ∈ (0...𝑦))
171	170	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑖 + 1) ∈ (0...𝑦))
172		fvres 6787	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑖 + 1) ∈ (0...𝑦) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
173	171, 172	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
174	173	adantlr 712	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → ((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)) = (𝑝‘(𝑖 + 1)))
175	169, 174	oveq12d 7287	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) = ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))
176	175	eleq2d 2824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) ∧ 𝑖 ∈ (0..^𝑦)) → (𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) ↔ 𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
177	176	rexbidva 3224	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) ↔ ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
178		nnz 12331	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
179		uzid 12586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ ℤ → 𝑦 ∈ (ℤ_≥‘𝑦))
180	178, 179	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ (ℤ_≥‘𝑦))
181		peano2uz 12630	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 ∈ (ℤ_≥‘𝑦) → (𝑦 + 1) ∈ (ℤ_≥‘𝑦))
182		fzoss2 13404	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 + 1) ∈ (ℤ_≥‘𝑦) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
183	180, 181, 182	3syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℕ → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
184	183	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) → (0..^𝑦) ⊆ (0..^(𝑦 + 1)))
185		ssrexv 3989	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0..^𝑦) ⊆ (0..^(𝑦 + 1)) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
186	184, 185	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
187	177, 186	sylbid 239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) → (∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
188	165, 187	embantd 59	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ 𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦))) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
189	188	ex 413	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
190	189	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
191	152, 190	syld 47	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) ∧ ¬ (𝑝‘𝑦) ≤ 𝑋) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
192	191	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
193	192	com34 91	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
194	193	com13 88	. . . . . . . . . . . . 13 ⊢ ((𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘0)[,)((𝑝 ↾ (0...𝑦))‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ (((𝑝 ↾ (0...𝑦))‘𝑖)[,)((𝑝 ↾ (0...𝑦))‘(𝑖 + 1)))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
195	135, 194	sylbi 216	. . . . . . . . . . . 12 ⊢ ([(𝑝 ↾ (0...𝑦)) / 𝑝](𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
196	97, 195	syl 17	. . . . . . . . . . 11 ⊢ (((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
197	196	ex 413	. . . . . . . . . 10 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))))
198	197	com24 95	. . . . . . . . 9 ⊢ ((𝑝 ↾ (0...𝑦)) ∈ (RePart‘𝑦) → ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))))
199	96, 198	mpcom 38	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑝 ∈ (RePart‘(𝑦 + 1))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
200	199	ex 413	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (¬ (𝑝‘𝑦) ≤ 𝑋 → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))))
201	200	com24 95	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))))
202	201	imp 407	. . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → (¬ (𝑝‘𝑦) ≤ 𝑋 → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))))
203	95, 202	pm2.61d 179	. . . 4 ⊢ ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → (𝑝 ∈ (RePart‘(𝑦 + 1)) → (𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
204	46, 203	ralrimi 3141	. . 3 ⊢ ((𝑦 ∈ ℕ ∧ ∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))) → ∀𝑝 ∈ (RePart‘(𝑦 + 1))(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))
205	204	ex 413	. 2 ⊢ (𝑦 ∈ ℕ → (∀𝑝 ∈ (RePart‘𝑦)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑦)) → ∃𝑖 ∈ (0..^𝑦)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))) → ∀𝑝 ∈ (RePart‘(𝑦 + 1))(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘(𝑦 + 1))) → ∃𝑖 ∈ (0..^(𝑦 + 1))𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1))))))
206	10, 18, 26, 34, 43, 205	nnind 11980	1 ⊢ (𝑀 ∈ ℕ → ∀𝑝 ∈ (RePart‘𝑀)(𝑋 ∈ ((𝑝‘0)[,)(𝑝‘𝑀)) → ∃𝑖 ∈ (0..^𝑀)𝑋 ∈ ((𝑝‘𝑖)[,)(𝑝‘(𝑖 + 1)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3431 [wsbc 3717 ⦋csb 3833 ⊆ wss 3888 {csn 4563 class class class wbr 5075 ↾ cres 5588 ‘cfv 6428 (class class class)co 7269 0cc0 10860 1c1 10861 + caddc 10863 ℝ^*cxr 10997 < clt 10998 ≤ cle 10999 ℕcn 11962 ℕ₀cn0 12222 ℤcz 12308 ℤ_≥cuz 12571 [,)cico 13070 ...cfz 13228 ..^cfzo 13371 RePartciccp 44822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-er 8487 df-map 8606 df-en 8723 df-dom 8724 df-sdom 8725 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-n0 12223 df-z 12309 df-uz 12572 df-ico 13074 df-fz 13229 df-fzo 13372 df-iccp 44823

This theorem is referenced by: iccpartiun 44843 icceuelpart 44845 bgoldbtbnd 45218

W3C validator