stoweidlem14 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem14 38907

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 3650	. . . . . . 7 ⊢ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ⊆ ℕ
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ⊆ ℕ)
4	1, 3	syl5eqss 3612	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
5		stoweidlem14.2	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
6	5	rprecred 11759	. . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ)
7		arch 11166	. . . . . 6 ⊢ ((1 / 𝐷) ∈ ℝ → ∃𝑘 ∈ ℕ (1 / 𝐷) < 𝑘)
8		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < 𝑘))
9	8	elrab 3331	. . . . . . . . . 10 ⊢ (𝑘 ∈ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗} ↔ (𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘))
10	9	biimpri 217	. . . . . . . . 9 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → 𝑘 ∈ {𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗})
11	10, 1	syl6eleqr 2699	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → 𝑘 ∈ 𝐴)
12		simpr 476	. . . . . . . 8 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → (1 / 𝐷) < 𝑘)
13	11, 12	jca 553	. . . . . . 7 ⊢ ((𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘) → (𝑘 ∈ 𝐴 ∧ (1 / 𝐷) < 𝑘))
14	13	reximi2 2993	. . . . . 6 ⊢ (∃𝑘 ∈ ℕ (1 / 𝐷) < 𝑘 → ∃𝑘 ∈ 𝐴 (1 / 𝐷) < 𝑘)
15		rexn0 4026	. . . . . 6 ⊢ (∃𝑘 ∈ 𝐴 (1 / 𝐷) < 𝑘 → 𝐴 ≠ ∅)
16	6, 7, 14, 15	4syl 19	. . . . 5 ⊢ (𝜑 → 𝐴 ≠ ∅)
17		nnwo 11629	. . . . 5 ⊢ ((𝐴 ⊆ ℕ ∧ 𝐴 ≠ ∅) → ∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
18	4, 16, 17	syl2anc 691	. . . 4 ⊢ (𝜑 → ∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
19		df-rex 2902	. . . 4 ⊢ (∃𝑘 ∈ 𝐴 ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧 ↔ ∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
20	18, 19	sylib 207	. . 3 ⊢ (𝜑 → ∃𝑘(𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
21	8, 1	elrab2 3333	. . . . . . . 8 ⊢ (𝑘 ∈ 𝐴 ↔ (𝑘 ∈ ℕ ∧ (1 / 𝐷) < 𝑘))
22	21	simplbi 475	. . . . . . 7 ⊢ (𝑘 ∈ 𝐴 → 𝑘 ∈ ℕ)
23	22	ad2antrl 760	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝑘 ∈ ℕ)
24		simpl 472	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝜑)
25		simprl 790	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 𝑘 ∈ 𝐴)
26		simprr 792	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
27		nfcv 2751	. . . . . . . . 9 ⊢ Ⅎ𝑧𝐴
28		nfrab1 3099	. . . . . . . . . 10 ⊢ Ⅎ𝑗{𝑗 ∈ ℕ ∣ (1 / 𝐷) < 𝑗}
29	1, 28	nfcxfr 2749	. . . . . . . . 9 ⊢ Ⅎ𝑗𝐴
30		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑗 𝑘 ≤ 𝑧
31		nfv 1830	. . . . . . . . 9 ⊢ Ⅎ𝑧 𝑘 ≤ 𝑗
32		breq2 4587	. . . . . . . . 9 ⊢ (𝑧 = 𝑗 → (𝑘 ≤ 𝑧 ↔ 𝑘 ≤ 𝑗))
33	27, 29, 30, 31, 32	cbvralf 3141	. . . . . . . 8 ⊢ (∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧 ↔ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)
34	26, 33	sylib 207	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)
35	21	simprbi 479	. . . . . . . . 9 ⊢ (𝑘 ∈ 𝐴 → (1 / 𝐷) < 𝑘)
36	35	ad2antrl 760	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → (1 / 𝐷) < 𝑘)
37	22	ad2antrl 760	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → 𝑘 ∈ ℕ)
38		1red 9934	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 1 ∈ ℝ)
39		nnre 10904	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
40	39	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑘 ∈ ℝ)
41	5	rpregt0d 11754	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 ∈ ℝ ∧ 0 < 𝐷))
42	41	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐷 ∈ ℝ ∧ 0 < 𝐷))
43		ltdivmul2 10779	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 𝑘 ∈ ℝ ∧ (𝐷 ∈ ℝ ∧ 0 < 𝐷)) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷)))
44	38, 40, 42, 43	syl3anc 1318	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷)))
45	37, 44	syldan 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → ((1 / 𝐷) < 𝑘 ↔ 1 < (𝑘 · 𝐷)))
46	36, 45	mpbid 221	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑗 ∈ 𝐴 𝑘 ≤ 𝑗)) → 1 < (𝑘 · 𝐷))
47	24, 25, 34, 46	syl12anc 1316	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → 1 < (𝑘 · 𝐷))
48		oveq1 6556	. . . . . . . . . . . 12 ⊢ (𝑘 = 1 → (𝑘 · 𝐷) = (1 · 𝐷))
49	48	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 1) → (𝑘 · 𝐷) = (1 · 𝐷))
50	5	rpcnd 11750	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐷 ∈ ℂ)
51	50	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 = 1) → 𝐷 ∈ ℂ)
52	51	mulid2d 9937	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 = 1) → (1 · 𝐷) = 𝐷)
53	49, 52	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 = 1) → (𝑘 · 𝐷) = 𝐷)
54	53	oveq1d 6564	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) = (𝐷 / 2))
55	5	rpred 11748	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 ∈ ℝ)
56	55	rehalfcld 11156	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
57		halfre 11123	. . . . . . . . . . . 12 ⊢ (1 / 2) ∈ ℝ
58	57	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 2) ∈ ℝ)
59		1red 9934	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℝ)
60		stoweidlem14.3	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐷 < 1)
61		2re 10967	. . . . . . . . . . . . . 14 ⊢ 2 ∈ ℝ
62	61	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ∈ ℝ)
63		2pos 10989	. . . . . . . . . . . . . 14 ⊢ 0 < 2
64	63	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 0 < 2)
65		ltdiv1 10766	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ ℝ ∧ 1 ∈ ℝ ∧ (2 ∈ ℝ ∧ 0 < 2)) → (𝐷 < 1 ↔ (𝐷 / 2) < (1 / 2)))
66	55, 59, 62, 64, 65	syl112anc 1322	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷 < 1 ↔ (𝐷 / 2) < (1 / 2)))
67	60, 66	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 / 2) < (1 / 2))
68		halflt1 11127	. . . . . . . . . . . 12 ⊢ (1 / 2) < 1
69	68	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (1 / 2) < 1)
70	56, 58, 59, 67, 69	lttrd 10077	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 / 2) < 1)
71	70	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 = 1) → (𝐷 / 2) < 1)
72	54, 71	eqbrtrd 4605	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1)
73	72	adantlr 747	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1)
74		simpll 786	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝜑)
75		simplrl 796	. . . . . . . . . 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 2790	. . . . . . . . . 10 ⊢ (¬ 𝑘 = 1 → 𝑘 ≠ 1)
78	77	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ≠ 1)
79		eluz2b3 11638	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ (𝑘 ∈ ℕ ∧ 𝑘 ≠ 1))
80	76, 78, 79	sylanbrc 695	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝑘 ∈ (ℤ_≥‘2))
81		peano2rem 10227	. . . . . . . . . 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 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝐷 ∈ ℝ)
84	5	rpne0d 11753	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ≠ 0)
85	84	ad2antrr 758	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 𝐷 ≠ 0)
86	83, 85	rereccld 10731	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (1 / 𝐷) ∈ ℝ)
87		1zzd 11285	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → 1 ∈ ℤ)
88		df-2 10956	. . . . . . . . . . . . . . 15 ⊢ 2 = (1 + 1)
89	88	fveq2i 6106	. . . . . . . . . . . . . 14 ⊢ (ℤ_≥‘2) = (ℤ_≥‘(1 + 1))
90	89	eleq2i 2680	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘2) ↔ 𝑘 ∈ (ℤ_≥‘(1 + 1)))
91		eluzsub 11593	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ_≥‘(1 + 1))) → (𝑘 − 1) ∈ (ℤ_≥‘1))
92	90, 91	syl3an3b 1356	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 − 1) ∈ (ℤ_≥‘1))
93		nnuz 11599	. . . . . . . . . . . 12 ⊢ ℕ = (ℤ_≥‘1)
94	92, 93	syl6eleqr 2699	. . . . . . . . . . 11 ⊢ ((1 ∈ ℤ ∧ 1 ∈ ℤ ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 − 1) ∈ ℕ)
95	87, 87, 80, 94	syl3anc 1318	. . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ ℝ)
98	97, 81	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → (𝑘 − 1) ∈ ℝ)
99		simpr 476	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → 𝑘 ∈ ℝ)
100	99	ltm1d 10835	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑘 − 1) < 𝑘)
101		ltnle 9996	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ((𝑘 − 1) < 𝑘 ↔ ¬ 𝑘 ≤ (𝑘 − 1)))
102	100, 101	mpbid 221	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑘 − 1) ∈ ℝ ∧ 𝑘 ∈ ℝ) → ¬ 𝑘 ≤ (𝑘 − 1))
103	98, 97, 102	syl2anc 691	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ¬ 𝑘 ≤ (𝑘 − 1))
104		breq2 4587	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = (𝑘 − 1) → (𝑘 ≤ 𝑧 ↔ 𝑘 ≤ (𝑘 − 1)))
105	104	notbid 307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = (𝑘 − 1) → (¬ 𝑘 ≤ 𝑧 ↔ ¬ 𝑘 ≤ (𝑘 − 1)))
106	105	rspcev 3282	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ ¬ 𝑘 ≤ (𝑘 − 1)) → ∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧)
107	103, 106	syldan 486	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧)
108		rexnal 2978	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑧 ∈ 𝐴 ¬ 𝑘 ≤ 𝑧 ↔ ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
109	107, 108	sylib 207	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑘 − 1) ∈ 𝐴 ∧ 𝑘 ∈ 𝐴) → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)
110	109	ex 449	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 − 1) ∈ 𝐴 → (𝑘 ∈ 𝐴 → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
111		imnan 437	. . . . . . . . . . . . . . . 16 ⊢ ((𝑘 ∈ 𝐴 → ¬ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) ↔ ¬ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
112	110, 111	sylib 207	. . . . . . . . . . . . . . 15 ⊢ ((𝑘 − 1) ∈ 𝐴 → ¬ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧))
113	112	con2i 133	. . . . . . . . . . . . . 14 ⊢ ((𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → ¬ (𝑘 − 1) ∈ 𝐴)
114	113	ad2antlr 759	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ (𝑘 − 1) ∈ 𝐴)
115		breq2 4587	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑘 − 1) → ((1 / 𝐷) < 𝑗 ↔ (1 / 𝐷) < (𝑘 − 1)))
116	115, 1	elrab2 3333	. . . . . . . . . . . . 13 ⊢ ((𝑘 − 1) ∈ 𝐴 ↔ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1)))
117	114, 116	sylnib 317	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ¬ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1)))
118		ianor 508	. . . . . . . . . . . 12 ⊢ (¬ ((𝑘 − 1) ∈ ℕ ∧ (1 / 𝐷) < (𝑘 − 1)) ↔ (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 / 𝐷) < (𝑘 − 1)))
119	117, 118	sylib 207	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 / 𝐷) < (𝑘 − 1)))
120		imor 427	. . . . . . . . . . 11 ⊢ (((𝑘 − 1) ∈ ℕ → ¬ (1 / 𝐷) < (𝑘 − 1)) ↔ (¬ (𝑘 − 1) ∈ ℕ ∨ ¬ (1 / 𝐷) < (𝑘 − 1)))
121	119, 120	sylibr 223	. . . . . . . . . 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 10066	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → (𝑘 − 1) ≤ (1 / 𝐷))
124		eluzelre 11574	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℝ)
125	124	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝑘 ∈ ℝ)
126	55	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝐷 ∈ ℝ)
127	125, 126	remulcld 9949	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 · 𝐷) ∈ ℝ)
128	127	rehalfcld 11156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 · 𝐷) / 2) ∈ ℝ)
129	128	3adant3 1074	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) / 2) ∈ ℝ)
130	59, 55	readdcld 9948	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐷) ∈ ℝ)
131	130	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (1 + 𝐷) ∈ ℝ)
132	131	rehalfcld 11156	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((1 + 𝐷) / 2) ∈ ℝ)
133	132	3adant3 1074	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((1 + 𝐷) / 2) ∈ ℝ)
134		1red 9934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → 1 ∈ ℝ)
135		eluzelcn 11575	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 𝑘 ∈ ℂ)
136	135	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝑘 ∈ ℂ)
137	50	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 𝐷 ∈ ℂ)
138	136, 137	mulcld 9939	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (𝑘 · 𝐷) ∈ ℂ)
139	138	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 · 𝐷) ∈ ℂ)
140	50	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → 𝐷 ∈ ℂ)
141	139, 140	npcand 10275	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (((𝑘 · 𝐷) − 𝐷) + 𝐷) = (𝑘 · 𝐷))
142	127, 126	resubcld 10337	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 · 𝐷) − 𝐷) ∈ ℝ)
143	142	3adant3 1074	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) − 𝐷) ∈ ℝ)
144	55	3ad2ant1 1075	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → 𝐷 ∈ ℝ)
145		simp3 1056	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 − 1) ≤ (1 / 𝐷))
146		1red 9934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ (ℤ_≥‘2) → 1 ∈ ℝ)
147	124, 146	resubcld 10337	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (ℤ_≥‘2) → (𝑘 − 1) ∈ ℝ)
148	147	3ad2ant2 1076	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 − 1) ∈ ℝ)
149	6	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (1 / 𝐷) ∈ ℝ)
150	41	3ad2ant1 1075	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝐷 ∈ ℝ ∧ 0 < 𝐷))
151		lemul1 10754	. . . . . . . . . . . . . . 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 1318	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 − 1) ≤ (1 / 𝐷) ↔ ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷)))
153	145, 152	mpbid 221	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 − 1) · 𝐷) ≤ ((1 / 𝐷) · 𝐷))
154		1cnd 9935	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → 1 ∈ ℂ)
155	136, 154, 137	subdird 10366	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − (1 · 𝐷)))
156	137	mulid2d 9937	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → (1 · 𝐷) = 𝐷)
157	156	oveq2d 6565	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 · 𝐷) − (1 · 𝐷)) = ((𝑘 · 𝐷) − 𝐷))
158	155, 157	eqtrd 2644	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − 𝐷))
159	158	3adant3 1074	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 − 1) · 𝐷) = ((𝑘 · 𝐷) − 𝐷))
160		1cnd 9935	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 1 ∈ ℂ)
161	160, 50, 84	3jca 1235	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (1 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))
162	161	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (1 ∈ ℂ ∧ 𝐷 ∈ ℂ ∧ 𝐷 ≠ 0))
163		divcan1 10573	. . . . . . . . . . . . . 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 4614	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) − 𝐷) ≤ 1)
166	143, 134, 144, 165	leadd1dd 10520	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (((𝑘 · 𝐷) − 𝐷) + 𝐷) ≤ (1 + 𝐷))
167	141, 166	eqbrtrrd 4607	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 · 𝐷) ≤ (1 + 𝐷))
168	127	3adant3 1074	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (𝑘 · 𝐷) ∈ ℝ)
169	130	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (1 + 𝐷) ∈ ℝ)
170	61, 63	pm3.2i 470	. . . . . . . . . . . 12 ⊢ (2 ∈ ℝ ∧ 0 < 2)
171	170	a1i 11	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → (2 ∈ ℝ ∧ 0 < 2))
172		lediv1 10767	. . . . . . . . . . 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 1318	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) ≤ (1 + 𝐷) ↔ ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2)))
174	167, 173	mpbid 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) / 2) ≤ ((1 + 𝐷) / 2))
175	55, 59, 59, 60	ltadd2dd 10075	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 + 𝐷) < (1 + 1))
176		1p1e2 11011	. . . . . . . . . . . . 13 ⊢ (1 + 1) = 2
177	175, 176	syl6breq 4624	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 + 𝐷) < 2)
178		ltdiv1 10766	. . . . . . . . . . . . 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 1322	. . . . . . . . . . . 12 ⊢ (𝜑 → ((1 + 𝐷) < 2 ↔ ((1 + 𝐷) / 2) < (2 / 2)))
180	177, 179	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → ((1 + 𝐷) / 2) < (2 / 2))
181		2div2e1 11027	. . . . . . . . . . 11 ⊢ (2 / 2) = 1
182	180, 181	syl6breq 4624	. . . . . . . . . 10 ⊢ (𝜑 → ((1 + 𝐷) / 2) < 1)
183	182	3ad2ant1 1075	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((1 + 𝐷) / 2) < 1)
184	129, 133, 134, 174, 183	lelttrd 10074	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘2) ∧ (𝑘 − 1) ≤ (1 / 𝐷)) → ((𝑘 · 𝐷) / 2) < 1)
185	74, 80, 123, 184	syl3anc 1318	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) ∧ ¬ 𝑘 = 1) → ((𝑘 · 𝐷) / 2) < 1)
186	73, 185	pm2.61dan 828	. . . . . 6 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → ((𝑘 · 𝐷) / 2) < 1)
187	23, 47, 186	jca32 556	. . . . 5 ⊢ ((𝜑 ∧ (𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧)) → (𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))
188	187	ex 449	. . . 4 ⊢ (𝜑 → ((𝑘 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐴 𝑘 ≤ 𝑧) → (𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))))
189	188	eximdv 1833	. . 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 2902	. 2 ⊢ (∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1) ↔ ∃𝑘(𝑘 ∈ ℕ ∧ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1)))
192	190, 191	sylibr 223	1 ⊢ (𝜑 → ∃𝑘 ∈ ℕ (1 < (𝑘 · 𝐷) ∧ ((𝑘 · 𝐷) / 2) < 1))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ⊆ wss 3540 ∅c0 3874 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 ℤcz 11254 ℤ_≥cuz 11563 ℝ⁺crp 11708

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709

This theorem is referenced by: stoweidlem49 38942

W3C validator