ormkglobd - Mathbox for Ender Ting

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Theorem ormkglobd 46880

Description: If all adjacent elements of a certain sequence are ordered according to a relation which is a total order on S, then any element is so related to anything to right of it (so-called "global monotonicity"). Deduction form. (Contributed by Ender Ting, 30-Apr-2025.)

**Hypotheses**
Ref	Expression
ormkglobd.1	⊢ (𝜑 → 𝑅 Or 𝑆)
ormkglobd.2	⊢ (𝜑 → ∀𝑘 ∈ (0..^(𝑇 + 1))(𝐵‘𝑘) ∈ 𝑆)
ormkglobd.3	⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1)))

**Assertion**
Ref	Expression
ormkglobd	⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡)))

Distinct variable groups: 𝐵,𝑘 𝑅,𝑘 𝑆,𝑘 𝑇,𝑘,𝑡 𝜑,𝑘,𝑡 Allowed substitution hints: 𝐵(𝑡) 𝑅(𝑡) 𝑆(𝑡)

Proof of Theorem ormkglobd Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		2a1 28	. . . . . . 7 ⊢ (𝜑 → ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → 𝜑)))
2	1	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1)))) → (𝑘 < 𝑡 → 𝜑))
3		2a1 28	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑡 ∈ (1..^(𝑇 + 1)) → (𝑘 < 𝑡 → 𝑘 ∈ (0..^𝑇))))
4	3	imp 406	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → 𝑘 ∈ (0..^𝑇)))
5	4	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1)))) → (𝑘 < 𝑡 → 𝑘 ∈ (0..^𝑇)))
6	2, 5	jcad 512	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1)))) → (𝑘 < 𝑡 → (𝜑 ∧ 𝑘 ∈ (0..^𝑇))))
7		elfzoelz 13627	. . . . . . . . 9 ⊢ (𝑡 ∈ (1..^(𝑇 + 1)) → 𝑡 ∈ ℤ)
8	7	adantl 481	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑡 ∈ ℤ)
9	8	a1d 25	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → 𝑡 ∈ ℤ))
10		elfzoelz 13627	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
11	10	adantr 480	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑘 ∈ ℤ)
12		elfzoelz 13627	. . . . . . . . . 10 ⊢ (𝑡 ∈ (1..^(𝑇 + 1)) → 𝑡 ∈ ℤ)
13	12	adantl 481	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑡 ∈ ℤ)
14		zltp1le 12590	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℤ ∧ 𝑡 ∈ ℤ) → (𝑘 < 𝑡 ↔ (𝑘 + 1) ≤ 𝑡))
15	11, 13, 14	syl2anc 584	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 ↔ (𝑘 + 1) ≤ 𝑡))
16	15	biimpd 229	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → (𝑘 + 1) ≤ 𝑡))
17		elfzoelz 13627	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ (1..^(𝑇 + 1)) → 𝑡 ∈ ℤ)
18	17	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑡 ∈ ℤ)
19	18	zred 12645	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑡 ∈ ℝ)
20		elfzoel2 13626	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
21	20	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑇 ∈ ℤ)
22	21	zred 12645	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑇 ∈ ℝ)
23		1red 11182	. . . . . . . . . 10 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 1 ∈ ℝ)
24	19, 22, 23	3jca 1128	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑡 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 1 ∈ ℝ))
25		elfzop1le2 13640	. . . . . . . . . 10 ⊢ (𝑡 ∈ (1..^(𝑇 + 1)) → (𝑡 + 1) ≤ (𝑇 + 1))
26	25	adantl 481	. . . . . . . . 9 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑡 + 1) ≤ (𝑇 + 1))
27		leadd1 11653	. . . . . . . . . 10 ⊢ ((𝑡 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → (𝑡 ≤ 𝑇 ↔ (𝑡 + 1) ≤ (𝑇 + 1)))
28	27	biimprd 248	. . . . . . . . 9 ⊢ ((𝑡 ∈ ℝ ∧ 𝑇 ∈ ℝ ∧ 1 ∈ ℝ) → ((𝑡 + 1) ≤ (𝑇 + 1) → 𝑡 ≤ 𝑇))
29	24, 26, 28	sylc 65	. . . . . . . 8 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → 𝑡 ≤ 𝑇)
30	29	a1d 25	. . . . . . 7 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → 𝑡 ≤ 𝑇))
31	9, 16, 30	3jcad 1129	. . . . . 6 ⊢ ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
32	31	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1)))) → (𝑘 < 𝑡 → (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))
33	6, 32	jcad 512	. . . 4 ⊢ ((𝜑 ∧ (𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1)))) → (𝑘 < 𝑡 → ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇))))
34	33	ex 412	. . 3 ⊢ (𝜑 → ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)))))
35		fveq2 6861	. . . . 5 ⊢ (𝑎 = (𝑘 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑘 + 1)))
36	35	breq2d 5122	. . . 4 ⊢ (𝑎 = (𝑘 + 1) → ((𝐵‘𝑘)𝑅(𝐵‘𝑎) ↔ (𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1))))
37		fveq2 6861	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐵‘𝑎) = (𝐵‘𝑏))
38	37	breq2d 5122	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐵‘𝑘)𝑅(𝐵‘𝑎) ↔ (𝐵‘𝑘)𝑅(𝐵‘𝑏)))
39		fveq2 6861	. . . . 5 ⊢ (𝑎 = (𝑏 + 1) → (𝐵‘𝑎) = (𝐵‘(𝑏 + 1)))
40	39	breq2d 5122	. . . 4 ⊢ (𝑎 = (𝑏 + 1) → ((𝐵‘𝑘)𝑅(𝐵‘𝑎) ↔ (𝐵‘𝑘)𝑅(𝐵‘(𝑏 + 1))))
41		fveq2 6861	. . . . 5 ⊢ (𝑎 = 𝑡 → (𝐵‘𝑎) = (𝐵‘𝑡))
42	41	breq2d 5122	. . . 4 ⊢ (𝑎 = 𝑡 → ((𝐵‘𝑘)𝑅(𝐵‘𝑎) ↔ (𝐵‘𝑘)𝑅(𝐵‘𝑡)))
43		ormkglobd.3	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)(𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1)))
44	43	r19.21bi 3230	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → (𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1)))
45		simp1l 1198	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝜑)
46		ormkglobd.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝑆)
47	45, 46	syl 17	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑅 Or 𝑆)
48		elfzofz 13643	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ (0...𝑇))
49		elfzoel2 13626	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
50		fzval3 13702	. . . . . . . . 9 ⊢ (𝑇 ∈ ℤ → (0...𝑇) = (0..^(𝑇 + 1)))
51	49, 50	syl 17	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑇) → (0...𝑇) = (0..^(𝑇 + 1)))
52	48, 51	eleqtrd 2831	. . . . . . 7 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ (0..^(𝑇 + 1)))
53		ormkglobd.2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑘 ∈ (0..^(𝑇 + 1))(𝐵‘𝑘) ∈ 𝑆)
54	53	r19.21bi 3230	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) → (𝐵‘𝑘) ∈ 𝑆)
55	52, 54	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → (𝐵‘𝑘) ∈ 𝑆)
56	55	3ad2ant1 1133	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝐵‘𝑘) ∈ 𝑆)
57		simp1l 1198	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝜑)
58		simp21 1207	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
59		0red 11184	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ∈ ℝ)
60		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
61		elfzoelz 13627	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
62	60, 61	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℤ)
63	62	zred 12645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℝ)
64		1red 11182	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
65	63, 64	readdcld 11210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑘 + 1) ∈ ℝ)
66		simp21 1207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
67	66	zred 12645	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℝ)
68		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
69		elfzoelz 13627	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
70	68, 69	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℤ)
71	70	zred 12645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℝ)
72		1red 11182	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
73		simp1r 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
74		elfzole1 13635	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
75	73, 74	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 𝑘)
76		0le1 11708	. . . . . . . . . . 11 ⊢ 0 ≤ 1
77	76	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 1)
78	71, 72, 75, 77	addge0d 11761	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
79		simp22 1208	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑘 + 1) ≤ 𝑏)
80	59, 65, 67, 78, 79	letrd 11338	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 𝑏)
81		elnn0z 12549	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℤ ∧ 0 ≤ 𝑏))
82	58, 80, 81	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℕ₀)
83		simp1r 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
84		elfzoel2 13626	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
85	83, 84	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
86	85	peano2zd 12648	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑇 + 1) ∈ ℤ)
87		simp21 1207	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
88	87	zred 12645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℝ)
89		simp1r 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
90		elfzoel2 13626	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
91	89, 90	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
92	91	zred 12645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℝ)
93		simp1r 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
94		elfzoel2 13626	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
95	93, 94	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
96	95	zred 12645	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℝ)
97		1red 11182	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
98	96, 97	readdcld 11210	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑇 + 1) ∈ ℝ)
99		simp23 1209	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 < 𝑇)
100		simp1r 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
101		elfzoel2 13626	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
102	100, 101	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
103	102	zred 12645	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℝ)
104	103	ltp1d 12120	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 < (𝑇 + 1))
105	88, 92, 98, 99, 104	lttrd 11342	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 < (𝑇 + 1))
106		elfzo0z 13669	. . . . . . 7 ⊢ (𝑏 ∈ (0..^(𝑇 + 1)) ↔ (𝑏 ∈ ℕ₀ ∧ (𝑇 + 1) ∈ ℤ ∧ 𝑏 < (𝑇 + 1)))
107	82, 86, 105, 106	syl3anbrc 1344	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ (0..^(𝑇 + 1)))
108		eleq1w 2812	. . . . . . . . 9 ⊢ (𝑘 = 𝑏 → (𝑘 ∈ (0..^(𝑇 + 1)) ↔ 𝑏 ∈ (0..^(𝑇 + 1))))
109	108	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = 𝑏 → ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) ↔ (𝜑 ∧ 𝑏 ∈ (0..^(𝑇 + 1)))))
110	53	r19.21bi 3230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) → (𝐵‘𝑘) ∈ 𝑆)
111		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑘 = 𝑏 → (𝐵‘𝑘) = (𝐵‘𝑏))
112	111	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑘 = 𝑏 → ((𝐵‘𝑘) ∈ 𝑆 ↔ (𝐵‘𝑏) ∈ 𝑆))
113	110, 112	imbitrid 244	. . . . . . . 8 ⊢ (𝑘 = 𝑏 → ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) → (𝐵‘𝑏) ∈ 𝑆))
114	109, 113	sylbird 260	. . . . . . 7 ⊢ (𝑘 = 𝑏 → ((𝜑 ∧ 𝑏 ∈ (0..^(𝑇 + 1))) → (𝐵‘𝑏) ∈ 𝑆))
115		ax6ev 1969	. . . . . . 7 ⊢ ∃𝑘 𝑘 = 𝑏
116	114, 115	exlimiiv 1931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (0..^(𝑇 + 1))) → (𝐵‘𝑏) ∈ 𝑆)
117	57, 107, 116	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝐵‘𝑏) ∈ 𝑆)
118		simp1l 1198	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝜑)
119		simp21 1207	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
120		0red 11184	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ∈ ℝ)
121		simp1r 1199	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
122		elfzoelz 13627	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
123	121, 122	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℤ)
124	123	zred 12645	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℝ)
125		1red 11182	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
126	124, 125	readdcld 11210	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑘 + 1) ∈ ℝ)
127		simp21 1207	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
128	127	zred 12645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℝ)
129		simp1r 1199	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
130		elfzoelz 13627	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
131	129, 130	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℤ)
132	131	zred 12645	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℝ)
133		1red 11182	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
134		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
135		elfzole1 13635	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
136	134, 135	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 𝑘)
137	76	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 1)
138	132, 133, 136, 137	addge0d 11761	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
139		simp22 1208	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑘 + 1) ≤ 𝑏)
140	120, 126, 128, 138, 139	letrd 11338	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 𝑏)
141		elnn0z 12549	. . . . . . . . 9 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℤ ∧ 0 ≤ 𝑏))
142	119, 140, 141	sylanbrc 583	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℕ₀)
143		1nn0 12465	. . . . . . . . 9 ⊢ 1 ∈ ℕ₀
144	143	a1i 11	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℕ₀)
145	142, 144	nn0addcld 12514	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑏 + 1) ∈ ℕ₀)
146		simp1r 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
147		elfzoel2 13626	. . . . . . . . 9 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
148	146, 147	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
149	148	peano2zd 12648	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑇 + 1) ∈ ℤ)
150		simp21 1207	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
151	150	zred 12645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℝ)
152		simp1r 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
153		elfzoel2 13626	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
154	152, 153	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
155	154	zred 12645	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℝ)
156		1red 11182	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
157		simp23 1209	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 < 𝑇)
158	151, 155, 156, 157	ltadd1dd 11796	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑏 + 1) < (𝑇 + 1))
159		elfzo0z 13669	. . . . . . 7 ⊢ ((𝑏 + 1) ∈ (0..^(𝑇 + 1)) ↔ ((𝑏 + 1) ∈ ℕ₀ ∧ (𝑇 + 1) ∈ ℤ ∧ (𝑏 + 1) < (𝑇 + 1)))
160	145, 149, 158, 159	syl3anbrc 1344	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑏 + 1) ∈ (0..^(𝑇 + 1)))
161		ovex 7423	. . . . . . 7 ⊢ (𝑏 + 1) ∈ V
162		eleq1 2817	. . . . . . . . 9 ⊢ (𝑘 = (𝑏 + 1) → (𝑘 ∈ (0..^(𝑇 + 1)) ↔ (𝑏 + 1) ∈ (0..^(𝑇 + 1))))
163	162	anbi2d 630	. . . . . . . 8 ⊢ (𝑘 = (𝑏 + 1) → ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) ↔ (𝜑 ∧ (𝑏 + 1) ∈ (0..^(𝑇 + 1)))))
164	53	r19.21bi 3230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) → (𝐵‘𝑘) ∈ 𝑆)
165		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑘 = (𝑏 + 1) → (𝐵‘𝑘) = (𝐵‘(𝑏 + 1)))
166	165	eleq1d 2814	. . . . . . . . 9 ⊢ (𝑘 = (𝑏 + 1) → ((𝐵‘𝑘) ∈ 𝑆 ↔ (𝐵‘(𝑏 + 1)) ∈ 𝑆))
167	164, 166	imbitrid 244	. . . . . . . 8 ⊢ (𝑘 = (𝑏 + 1) → ((𝜑 ∧ 𝑘 ∈ (0..^(𝑇 + 1))) → (𝐵‘(𝑏 + 1)) ∈ 𝑆))
168	163, 167	sylbird 260	. . . . . . 7 ⊢ (𝑘 = (𝑏 + 1) → ((𝜑 ∧ (𝑏 + 1) ∈ (0..^(𝑇 + 1))) → (𝐵‘(𝑏 + 1)) ∈ 𝑆))
169	161, 168	vtocle 3524	. . . . . 6 ⊢ ((𝜑 ∧ (𝑏 + 1) ∈ (0..^(𝑇 + 1))) → (𝐵‘(𝑏 + 1)) ∈ 𝑆)
170	118, 160, 169	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝐵‘(𝑏 + 1)) ∈ 𝑆)
171		simp3 1138	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝐵‘𝑘)𝑅(𝐵‘𝑏))
172		simp1l 1198	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝜑)
173		simp21 1207	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
174		0red 11184	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ∈ ℝ)
175		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
176		elfzoelz 13627	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
177	175, 176	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℤ)
178	177	zred 12645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℝ)
179		1red 11182	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
180	178, 179	readdcld 11210	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑘 + 1) ∈ ℝ)
181		simp21 1207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℤ)
182	181	zred 12645	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℝ)
183		simp1r 1199	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
184		elfzoelz 13627	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
185	183, 184	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℤ)
186	185	zred 12645	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ ℝ)
187		1red 11182	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 1 ∈ ℝ)
188		simp1r 1199	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
189		elfzole1 13635	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0..^𝑇) → 0 ≤ 𝑘)
190	188, 189	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 𝑘)
191	76	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 1)
192	186, 187, 190, 191	addge0d 11761	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ (𝑘 + 1))
193		simp22 1208	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝑘 + 1) ≤ 𝑏)
194	174, 180, 182, 192, 193	letrd 11338	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 0 ≤ 𝑏)
195		elnn0z 12549	. . . . . . . 8 ⊢ (𝑏 ∈ ℕ₀ ↔ (𝑏 ∈ ℤ ∧ 0 ≤ 𝑏))
196	173, 194, 195	sylanbrc 583	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ ℕ₀)
197		simp1r 1199	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑘 ∈ (0..^𝑇))
198		elfzoel2 13626	. . . . . . . 8 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
199	197, 198	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑇 ∈ ℤ)
200		simp23 1209	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 < 𝑇)
201		elfzo0z 13669	. . . . . . 7 ⊢ (𝑏 ∈ (0..^𝑇) ↔ (𝑏 ∈ ℕ₀ ∧ 𝑇 ∈ ℤ ∧ 𝑏 < 𝑇))
202	196, 199, 200, 201	syl3anbrc 1344	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → 𝑏 ∈ (0..^𝑇))
203		eleq1w 2812	. . . . . . . . 9 ⊢ (𝑏 = 𝑘 → (𝑏 ∈ (0..^𝑇) ↔ 𝑘 ∈ (0..^𝑇)))
204	203	anbi2d 630	. . . . . . . 8 ⊢ (𝑏 = 𝑘 → ((𝜑 ∧ 𝑏 ∈ (0..^𝑇)) ↔ (𝜑 ∧ 𝑘 ∈ (0..^𝑇))))
205	43	r19.21bi 3230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → (𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1)))
206		fveq2 6861	. . . . . . . . . 10 ⊢ (𝑏 = 𝑘 → (𝐵‘𝑏) = (𝐵‘𝑘))
207		fvoveq1 7413	. . . . . . . . . 10 ⊢ (𝑏 = 𝑘 → (𝐵‘(𝑏 + 1)) = (𝐵‘(𝑘 + 1)))
208	206, 207	breq12d 5123	. . . . . . . . 9 ⊢ (𝑏 = 𝑘 → ((𝐵‘𝑏)𝑅(𝐵‘(𝑏 + 1)) ↔ (𝐵‘𝑘)𝑅(𝐵‘(𝑘 + 1))))
209	205, 208	imbitrrid 246	. . . . . . . 8 ⊢ (𝑏 = 𝑘 → ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → (𝐵‘𝑏)𝑅(𝐵‘(𝑏 + 1))))
210	204, 209	sylbid 240	. . . . . . 7 ⊢ (𝑏 = 𝑘 → ((𝜑 ∧ 𝑏 ∈ (0..^𝑇)) → (𝐵‘𝑏)𝑅(𝐵‘(𝑏 + 1))))
211		ax6evr 2015	. . . . . . 7 ⊢ ∃𝑘 𝑏 = 𝑘
212	210, 211	exlimiiv 1931	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (0..^𝑇)) → (𝐵‘𝑏)𝑅(𝐵‘(𝑏 + 1)))
213	172, 202, 212	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝐵‘𝑏)𝑅(𝐵‘(𝑏 + 1)))
214	47, 56, 117, 170, 171, 213	sotrd 5575	. . . 4 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑏 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑏 ∧ 𝑏 < 𝑇) ∧ (𝐵‘𝑘)𝑅(𝐵‘𝑏)) → (𝐵‘𝑘)𝑅(𝐵‘(𝑏 + 1)))
215		elfzoelz 13627	. . . . . 6 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑘 ∈ ℤ)
216	215	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → 𝑘 ∈ ℤ)
217	216	peano2zd 12648	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → (𝑘 + 1) ∈ ℤ)
218		elfzoel2 13626	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑇) → 𝑇 ∈ ℤ)
219	218	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → 𝑇 ∈ ℤ)
220		elfzop1le2 13640	. . . . 5 ⊢ (𝑘 ∈ (0..^𝑇) → (𝑘 + 1) ≤ 𝑇)
221	220	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) → (𝑘 + 1) ≤ 𝑇)
222	36, 38, 40, 42, 44, 214, 217, 219, 221	fzindd 12643	. . 3 ⊢ (((𝜑 ∧ 𝑘 ∈ (0..^𝑇)) ∧ (𝑡 ∈ ℤ ∧ (𝑘 + 1) ≤ 𝑡 ∧ 𝑡 ≤ 𝑇)) → (𝐵‘𝑘)𝑅(𝐵‘𝑡))
223	34, 222	syl8 76	. 2 ⊢ (𝜑 → ((𝑘 ∈ (0..^𝑇) ∧ 𝑡 ∈ (1..^(𝑇 + 1))) → (𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡))))
224	223	ralrimivv 3179	1 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝑇)∀𝑡 ∈ (1..^(𝑇 + 1))(𝑘 < 𝑡 → (𝐵‘𝑘)𝑅(𝐵‘𝑡)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 class class class wbr 5110 Or wor 5548 ‘cfv 6514 (class class class)co 7390 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 < clt 11215 ≤ cle 11216 ℕ₀cn0 12449 ℤcz 12536 ...cfz 13475 ..^cfzo 13622

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 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623

This theorem is referenced by: (None)

W3C validator