stoweidlem14 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem14 42293

Description: There exists a 𝑘 as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 𝑘 is an integer and 1 < k * δ < 2. 𝐷 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem14.1	⊢ 𝐴 = {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}
stoweidlem14.2	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
stoweidlem14.3	⊢ (𝜑 → 𝐷 < 1)

**Assertion**
Ref	Expression
stoweidlem14	⊢ (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))

Distinct variable groups: 𝑗,𝑘,𝐷 𝐴,𝑘 𝜑,𝑘 Allowed substitution hints: 𝜑(𝑗) 𝐴(𝑗)

Proof of Theorem stoweidlem14 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem14.1	. . . . . 6 ⊢ 𝐴 = {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}
2		ssrab2 4055	. . . . . . 7 ⊢ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ⊆ ℕ
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ⊆ ℕ)
4	1, 3	eqsstrid 4014	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
5		stoweidlem14.2	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
6	5	rprecred 12436	. . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ)
7		arch 11888	. . . . . 6 ⊢ ((1 / 𝐷) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / 𝐷) < 𝑘)
8		breq2 5062	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < 𝑘))
9	8	elrab 3679	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ↔ (𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘))
10	9	biimpri 230	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → 𝑘 ∈ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗})
11	10, 1	eleqtrrdi 2924	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → 𝑘 ∈ 𝐴)
12		simpr 487	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → (1 / 𝐷) < 𝑘)
13	11, 12	jca 514	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → (𝑘 ∈ 𝐴 ∧ (1 / 𝐷) < 𝑘))
14	13	reximi2 3244	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ (1 / 𝐷) < 𝑘 → ∃𝑘 ∈ 𝐴 (1 / 𝐷) < 𝑘)
15		rexn0 4453	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 (1 / 𝐷) < 𝑘 → 𝐴 ≠ ∅)
16	6, 7, 14, 15	4syl 19	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅)
17		nnwo 12307	. . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
18	4, 16, 17	syl2anc 586	. . . 4 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
19		df-rex 3144	. . . 4 ⊢ (∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧 ↔ ∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
20	18, 19	sylib 220	. . 3 ⊢ (𝜑 → ∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
21	8, 1	elrab2 3682	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘))
22	21	simplbi 500	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ ℕ)
23	22	ad2antrl 726	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝑘 ∈ ℕ)
24		simpl 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝜑)
25		simprl 769	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝑘 ∈ 𝐴)
26		simprr 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
27		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑧𝐴
28		nfrab1 3384	. . . . . . . . . 10 ⊢ Ⅎ𝑗{𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}
29	1, 28	nfcxfr 2975	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐴
30		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑧
31		nfv 1911	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑘 ≤ 𝑗
32		breq2 5062	. . . . . . . . 9 ⊢ (𝑧 = 𝑗 → (𝑘 ≤ 𝑧 ↔ 𝑘 ≤ 𝑗))
33	27, 29, 30, 31, 32	cbvralfw 3437	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧 ↔ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)
34	26, 33	sylib 220	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)
35	21	simprbi 499	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (1 / 𝐷) < 𝑘)
36	35	ad2antrl 726	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → (1 / 𝐷) < 𝑘)
37	22	ad2antrl 726	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → 𝑘 ∈ ℕ)
38		1red 10636	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
39		nnre 11639	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
40	39	adantl 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
41	5	rpregt0d 12431	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 < 𝐷))
42	41	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐷 ∈ ℝ ∧ 0 < 𝐷))
43		ltdivmul2 11511	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷)) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷)))
44	38, 40, 42, 43	syl3anc 1367	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷)))
45	37, 44	syldan 593	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷)))
46	36, 45	mpbid 234	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → 1 < (𝑘 · 𝐷))
47	24, 25, 34, 46	syl12anc 834	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 1 < (𝑘 · 𝐷))
48		oveq1 7157	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (𝑘 · 𝐷) = (1 · 𝐷))
49	48	adantl 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 1) → (𝑘 · 𝐷) = (1 · 𝐷))
50	5	rpcnd 12427	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℂ)
51	50	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
52	51	mulid2d 10653	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 1) → (1 · 𝐷) = 𝐷)
53	49, 52	eqtrd 2856	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 1) → (𝑘 · 𝐷) = 𝐷)
54	53	oveq1d 7165	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) = (𝐷 / 2))
55	5	rpred 12425	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ ℝ)
56	55	rehalfcld 11878	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
57		halfre 11845	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
58	57	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
59		1red 10636	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
60		stoweidlem14.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 < 1)
61		2re 11705	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
63		2pos 11734	. . . . . . . . . . . . . 14 ⊢ 0 < 2
64	63	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 2)
65		ltdiv1 11498	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ ℝ ∧ 1 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝐷 < 1 ↔ (𝐷 / 2) < (1 / 2)))
66	55, 59, 62, 64, 65	syl112anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 < 1 ↔ (𝐷 / 2) < (1 / 2)))
67	60, 66	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 / 2) < (1 / 2))
68		halflt1 11849	. . . . . . . . . . . 12 ⊢ (1 / 2) < 1
69	68	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 2) < 1)
70	56, 58, 59, 67, 69	lttrd 10795	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 / 2) < 1)
71	70	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 1) → (𝐷 / 2) < 1)
72	54, 71	eqbrtrd 5080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1)
73	72	adantlr 713	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1)
74		simpll 765	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝜑)
75		simplrl 775	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ 𝐴)
76	75, 22	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ ℕ)
77		neqne 3024	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 1 → 𝑘 ≠ 1)
78	77	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ≠ 1)
79		eluz2b3 12316	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
80	76, 78, 79	sylanbrc 585	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ (ℤ_≥‘2))
81		peano2rem 10947	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℝ → (𝑘 − 1) ∈ ℝ)
82	75, 22, 39, 81	4syl 19	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℝ)
83	55	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝐷 ∈ ℝ)
84	5	rpne0d 12430	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 0)
85	84	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝐷 ≠ 0)
86	83, 85	rereccld 11461	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (1 / 𝐷) ∈ ℝ)
87		1zzd 12007	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℤ)
88		df-2 11694	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
89	88	fveq2i 6667	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
90	89	eleq2i 2904	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ 𝑘 ∈ (ℤ_≥‘(1 + 1)))
91		eluzsub 12268	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ_≥‘(1 + 1))) → (𝑘 − 1) ∈ (ℤ_≥‘1))
92	90, 91	syl3an3b 1401	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 − 1) ∈ (ℤ_≥‘1))
93		nnuz 12275	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
94	92, 93	eleqtrrdi 2924	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 − 1) ∈ ℕ)
95	87, 87, 80, 94	syl3anc 1367	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ∈ ℕ)
96	22, 39	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ ℝ)
97	96	adantl 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ)
98	97, 81	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑘 − 1) ∈ ℝ)
99		simpr 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
100	99	ltm1d 11566	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 − 1) < 𝑘)
101		ltnle 10714	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝑘 − 1) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑘 − 1)))
102	100, 101	mpbid 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ¬ 𝑘 ≤ (𝑘 − 1))
103	98, 97, 102	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ≤ (𝑘 − 1))
104		breq2 5062	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝑘 − 1) → (𝑘 ≤ 𝑧 ↔ 𝑘 ≤ (𝑘 − 1)))
105	104	notbid 320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑘 − 1) → (¬ 𝑘 ≤ 𝑧 ↔ ¬ 𝑘 ≤ (𝑘 − 1)))
106	105	rspcev 3622	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ ¬ 𝑘 ≤ (𝑘 − 1)) → ∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧)
107	103, 106	syldan 593	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧)
108		rexnal 3238	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
109	107, 108	sylib 220	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
110	109	ex 415	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 − 1) ∈ 𝐴 → (𝑘 ∈ 𝐴 → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
111		imnan 402	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝐴 → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) ↔ ¬ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
112	110, 111	sylib 220	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 − 1) ∈ 𝐴 → ¬ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
113	112	con2i 141	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → ¬ (𝑘 − 1) ∈ 𝐴)
114	113	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ (𝑘 − 1) ∈ 𝐴)
115		breq2 5062	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑘 − 1) → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < (𝑘 − 1)))
116	115, 1	elrab2 3682	. . . . . . . . . . . . 13 ⊢ ((𝑘 − 1) ∈ 𝐴 ↔ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1)))
117	114, 116	sylnib 330	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1)))
118		ianor 978	. . . . . . . . . . . 12 ⊢ (¬ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1)) ↔ (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 / 𝐷) < (𝑘 − 1)))
119	117, 118	sylib 220	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 / 𝐷) < (𝑘 − 1)))
120		imor 849	. . . . . . . . . . 11 ⊢ (((𝑘 − 1) ∈ ℕ → ¬ (1 / 𝐷) < (𝑘 − 1)) ↔ (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 / 𝐷) < (𝑘 − 1)))
121	119, 120	sylibr 236	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ((𝑘 − 1) ∈ ℕ → ¬ (1 / 𝐷) < (𝑘 − 1)))
122	95, 121	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ (1 / 𝐷) < (𝑘 − 1))
123	82, 86, 122	nltled 10784	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ≤ (1 / 𝐷))
124		eluzelre 12248	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℝ)
125	124	adantl 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝑘 ∈ ℝ)
126	55	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝐷 ∈ ℝ)
127	125, 126	remulcld 10665	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 · 𝐷) ∈ ℝ)
128	127	rehalfcld 11878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 · 𝐷) / 2) ∈ ℝ)
129	128	3adant3 1128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) / 2) ∈ ℝ)
130	59, 55	readdcld 10664	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐷) ∈ ℝ)
131	130	adantr 483	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (1 + 𝐷) ∈ ℝ)
132	131	rehalfcld 11878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((1 + 𝐷) / 2) ∈ ℝ)
133	132	3adant3 1128	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((1 + 𝐷) / 2) ∈ ℝ)
134		1red 10636	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → 1 ∈ ℝ)
135		eluzelcn 12249	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℂ)
136	135	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝑘 ∈ ℂ)
137	50	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝐷 ∈ ℂ)
138	136, 137	mulcld 10655	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 · 𝐷) ∈ ℂ)
139	138	3adant3 1128	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 · 𝐷) ∈ ℂ)
140	50	3ad2ant1 1129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → 𝐷 ∈ ℂ)
141	139, 140	npcand 10995	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (((𝑘 · 𝐷) − 𝐷) + 𝐷) = (𝑘 · 𝐷))
142	127, 126	resubcld 11062	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 · 𝐷) − 𝐷) ∈ ℝ)
143	142	3adant3 1128	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) − 𝐷) ∈ ℝ)
144	55	3ad2ant1 1129	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → 𝐷 ∈ ℝ)
145		simp3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 − 1) ≤ (1 / 𝐷))
146		1red 10636	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 1 ∈ ℝ)
147	124, 146	resubcld 11062	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (𝑘 − 1) ∈ ℝ)
148	147	3ad2ant2 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 − 1) ∈ ℝ)
149	6	3ad2ant1 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (1 / 𝐷) ∈ ℝ)
150	41	3ad2ant1 1129	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝐷 ∈ ℝ ∧ 0 < 𝐷))
151		lemul1 11486	. . . . . . . . . . . . . . 15 ⊢ (((𝑘 − 1) ∈ ℝ ∧ (1 / 𝐷) ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷)) → ((𝑘 − 1) ≤ (1 / 𝐷) ↔ ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷)))
152	148, 149, 150, 151	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 − 1) ≤ (1 / 𝐷) ↔ ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷)))
153	145, 152	mpbid 234	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷))
154		1cnd 10630	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 1 ∈ ℂ)
155	136, 154, 137	subdird 11091	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − (1 · 𝐷)))
156	137	mulid2d 10653	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (1 · 𝐷) = 𝐷)
157	156	oveq2d 7166	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 · 𝐷) − (1 · 𝐷)) = ((𝑘 · 𝐷) − 𝐷))
158	155, 157	eqtrd 2856	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − 𝐷))
159	158	3adant3 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − 𝐷))
160		1cnd 10630	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℂ)
161	160, 50, 84	3jca 1124	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))
162	161	3ad2ant1 1129	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (1 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))
163		divcan1 11301	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0) → ((1 / 𝐷) · 𝐷) = 1)
164	162, 163	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((1 / 𝐷) · 𝐷) = 1)
165	153, 159, 164	3brtr3d 5089	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) − 𝐷) ≤ 1)
166	143, 134, 144, 165	leadd1dd 11248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (((𝑘 · 𝐷) − 𝐷) + 𝐷) ≤ (1 + 𝐷))
167	141, 166	eqbrtrrd 5082	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 · 𝐷) ≤ (1 + 𝐷))
168	127	3adant3 1128	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 · 𝐷) ∈ ℝ)
169	130	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (1 + 𝐷) ∈ ℝ)
170	61, 63	pm3.2i 473	. . . . . . . . . . . 12 ⊢ (2 ∈ ℝ ∧ 0 < 2)
171	170	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (2 ∈ ℝ ∧ 0 < 2))
172		lediv1 11499	. . . . . . . . . . 11 ⊢ (((𝑘 · 𝐷) ∈ ℝ ∧ (1 + 𝐷) ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((𝑘 · 𝐷) ≤ (1 + 𝐷) ↔ ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2)))
173	168, 169, 171, 172	syl3anc 1367	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) ≤ (1 + 𝐷) ↔ ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2)))
174	167, 173	mpbid 234	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2))
175	55, 59, 59, 60	ltadd2dd 10793	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 + 𝐷) < (1 + 1))
176		1p1e2 11756	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
177	175, 176	breqtrdi 5099	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐷) < 2)
178		ltdiv1 11498	. . . . . . . . . . . . 13 ⊢ (((1 + 𝐷) ∈ ℝ ∧ 2 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → ((1 + 𝐷) < 2 ↔ ((1 + 𝐷) / 2) < (2 / 2)))
179	130, 62, 62, 64, 178	syl112anc 1370	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 + 𝐷) < 2 ↔ ((1 + 𝐷) / 2) < (2 / 2)))
180	177, 179	mpbid 234	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 + 𝐷) / 2) < (2 / 2))
181		2div2e1 11772	. . . . . . . . . . 11 ⊢ (2 / 2) = 1
182	180, 181	breqtrdi 5099	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐷) / 2) < 1)
183	182	3ad2ant1 1129	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((1 + 𝐷) / 2) < 1)
184	129, 133, 134, 174, 183	lelttrd 10792	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) / 2) < 1)
185	74, 80, 123, 184	syl3anc 1367	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1)
186	73, 185	pm2.61dan 811	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ((𝑘 · 𝐷) / 2) < 1)
187	23, 47, 186	jca32 518	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → (𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))
188	187	ex 415	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → (𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))))
189	188	eximdv 1914	. . 3 ⊢ (𝜑 → (∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))))
190	20, 189	mpd 15	. 2 ⊢ (𝜑 → ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))
191		df-rex 3144	. 2 ⊢ (∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1) ↔ ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))
192	190, 191	sylibr 236	1 ⊢ (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 {crab 3142 ⊆ wss 3935 ∅c0 4290 class class class wbr 5058 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℕcn 11632 2c2 11686 ℤcz 11975 ℤ_≥cuz 12237 ℝ⁺crp 12383

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384

This theorem is referenced by: stoweidlem49 42328

W3C validator