apdifflemf - Mathbox for Jim Kingdon

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Theorem apdifflemf 16761

Description: Lemma for apdiff 16763. Being apart from the point halfway between 𝑄 and 𝑅 suffices for 𝐴 to be a different distance from 𝑄 and from 𝑅. (Contributed by Jim Kingdon, 18-May-2024.)

**Hypotheses**
Ref	Expression
apdifflemf.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
apdifflemf.q	⊢ (𝜑 → 𝑄 ∈ ℚ)
apdifflemf.r	⊢ (𝜑 → 𝑅 ∈ ℚ)
apdifflemf.qr	⊢ (𝜑 → 𝑄 < 𝑅)
apdifflemf.ap	⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)

**Assertion**
Ref	Expression
apdifflemf	⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))

**Proof of Theorem apdifflemf**
Step	Hyp	Ref	Expression
1		apdifflemf.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2	1	recnd 8250	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ)
3		apdifflemf.r	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ ℚ)
4		qcn 9912	. . . . . . 7 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℂ)
5	3, 4	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ℂ)
6	2, 5	subcld 8532	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑅) ∈ ℂ)
7	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝐴 − 𝑅) ∈ ℂ)
8	7	abscld 11804	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
9		apdifflemf.q	. . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ ℚ)
10		qcn 9912	. . . . . . 7 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℂ)
11	9, 10	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑄 ∈ ℂ)
12	2, 11	subcld 8532	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝑄) ∈ ℂ)
13	12	abscld 11804	. . . 4 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
14	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
15		qre 9903	. . . . . . . . . 10 ⊢ (𝑄 ∈ ℚ → 𝑄 ∈ ℝ)
16	9, 15	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ℝ)
17	16	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ∈ ℝ)
18	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 ∈ ℝ)
19		qaddcl 9913	. . . . . . . . . . . . . 14 ⊢ ((𝑄 ∈ ℚ ∧ 𝑅 ∈ ℚ) → (𝑄 + 𝑅) ∈ ℚ)
20	9, 3, 19	syl2anc 411	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℚ)
21		qre 9903	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 𝑅) ∈ ℚ → (𝑄 + 𝑅) ∈ ℝ)
22	20, 21	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 𝑅) ∈ ℝ)
23	22	rehalfcld 9433	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) ∈ ℝ)
24	23	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) / 2) ∈ ℝ)
25		apdifflemf.qr	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑄 < 𝑅)
26		qre 9903	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ ℚ → 𝑅 ∈ ℝ)
27	3, 26	syl 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑅 ∈ ℝ)
28		avglt1 9425	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ 𝑄 < ((𝑄 + 𝑅) / 2)))
29	16, 27, 28	syl2anc 411	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 < 𝑅 ↔ 𝑄 < ((𝑄 + 𝑅) / 2)))
30	25, 29	mpbid 147	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 < ((𝑄 + 𝑅) / 2))
31	30	adantr 276	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < ((𝑄 + 𝑅) / 2))
32		simpr 110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) / 2) < 𝐴)
33	17, 24, 18, 31, 32	lttrd 8347	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝐴)
34	17, 18, 33	ltled 8340	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ≤ 𝐴)
35	17, 18, 34	abssubge0d 11799	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) = (𝐴 − 𝑄))
36	35	oveq2d 6044	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) = (𝑅 − (𝐴 − 𝑄)))
37	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑅 ∈ ℂ)
38	2	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 ∈ ℂ)
39	11	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 ∈ ℂ)
40	37, 38, 39	subsub3d 8562	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (𝐴 − 𝑄)) = ((𝑅 + 𝑄) − 𝐴))
41	37, 39	addcomd 8372	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + 𝑄) = (𝑄 + 𝑅))
42	41	oveq1d 6043	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑅 + 𝑄) − 𝐴) = ((𝑄 + 𝑅) − 𝐴))
43	36, 40, 42	3eqtrd 2268	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) = ((𝑄 + 𝑅) − 𝐴))
44	22	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) ∈ ℝ)
45		2rp 9937	. . . . . . . . . 10 ⊢ 2 ∈ ℝ⁺
46	45	a1i 9	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 2 ∈ ℝ⁺)
47	44, 18, 46	ltdivmuld 10027	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (((𝑄 + 𝑅) / 2) < 𝐴 ↔ (𝑄 + 𝑅) < (2 · 𝐴)))
48	32, 47	mpbid 147	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (2 · 𝐴))
49	38	2timesd 9429	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (2 · 𝐴) = (𝐴 + 𝐴))
50	48, 49	breqtrd 4119	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 + 𝑅) < (𝐴 + 𝐴))
51	44, 18, 18	ltsubaddd 8763	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (((𝑄 + 𝑅) − 𝐴) < 𝐴 ↔ (𝑄 + 𝑅) < (𝐴 + 𝐴)))
52	50, 51	mpbird 167	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((𝑄 + 𝑅) − 𝐴) < 𝐴)
53	43, 52	eqbrtrd 4115	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − (abs‘(𝐴 − 𝑄))) < 𝐴)
54	25	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑄 < 𝑅)
55	27	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝑅 ∈ ℝ)
56		difrp 9971	. . . . . . . 8 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ (𝑅 − 𝑄) ∈ ℝ⁺))
57	17, 55, 56	syl2anc 411	. . . . . . 7 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑄 < 𝑅 ↔ (𝑅 − 𝑄) ∈ ℝ⁺))
58	54, 57	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 − 𝑄) ∈ ℝ⁺)
59	18, 58	ltaddrpd 10009	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝐴 + (𝑅 − 𝑄)))
60	35	oveq2d 6044	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝑅 + (𝐴 − 𝑄)))
61	37, 38, 39	addsub12d 8555	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (𝐴 − 𝑄)) = (𝐴 + (𝑅 − 𝑄)))
62	60, 61	eqtrd 2264	. . . . 5 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (𝑅 + (abs‘(𝐴 − 𝑄))) = (𝐴 + (𝑅 − 𝑄)))
63	59, 62	breqtrrd 4121	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))
64	18, 55, 14	absdifltd 11801	. . . 4 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → ((abs‘(𝐴 − 𝑅)) < (abs‘(𝐴 − 𝑄)) ↔ ((𝑅 − (abs‘(𝐴 − 𝑄))) < 𝐴 ∧ 𝐴 < (𝑅 + (abs‘(𝐴 − 𝑄))))))
65	53, 63, 64	mpbir2and 953	. . 3 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑅)) < (abs‘(𝐴 − 𝑄)))
66	8, 14, 65	gtapd 8859	. 2 ⊢ ((𝜑 ∧ ((𝑄 + 𝑅) / 2) < 𝐴) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
67	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) ∈ ℝ)
68	6	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 − 𝑅) ∈ ℂ)
69	68	abscld 11804	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) ∈ ℝ)
70	11, 5, 2	subsubd 8560	. . . . . . 7 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) = ((𝑄 − 𝑅) + 𝐴))
71	16, 27	sublt0d 8792	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ 𝑄 < 𝑅))
72	25, 71	mpbird 167	. . . . . . . 8 ⊢ (𝜑 → (𝑄 − 𝑅) < 0)
73	16, 27	resubcld 8602	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 − 𝑅) ∈ ℝ)
74		ltaddnegr 8647	. . . . . . . . 9 ⊢ (((𝑄 − 𝑅) ∈ ℝ ∧ 𝐴 ∈ ℝ) → ((𝑄 − 𝑅) < 0 ↔ ((𝑄 − 𝑅) + 𝐴) < 𝐴))
75	73, 1, 74	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 − 𝑅) < 0 ↔ ((𝑄 − 𝑅) + 𝐴) < 𝐴))
76	72, 75	mpbid 147	. . . . . . 7 ⊢ (𝜑 → ((𝑄 − 𝑅) + 𝐴) < 𝐴)
77	70, 76	eqbrtrd 4115	. . . . . 6 ⊢ (𝜑 → (𝑄 − (𝑅 − 𝐴)) < 𝐴)
78	77	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑄 − (𝑅 − 𝐴)) < 𝐴)
79	1	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ∈ ℝ)
80	22	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑄 + 𝑅) ∈ ℝ)
81		simpr 110	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < ((𝑄 + 𝑅) / 2))
82	79, 79, 80, 81, 81	lt2halvesd 9434	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝐴 + 𝐴) < (𝑄 + 𝑅))
83	79, 79, 80	ltaddsub2d 8768	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝐴 + 𝐴) < (𝑄 + 𝑅) ↔ 𝐴 < ((𝑄 + 𝑅) − 𝐴)))
84	82, 83	mpbid 147	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < ((𝑄 + 𝑅) − 𝐴))
85	11	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑄 ∈ ℂ)
86	5	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑅 ∈ ℂ)
87	2	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ∈ ℂ)
88	85, 86, 87	addsubassd 8552	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) − 𝐴) = (𝑄 + (𝑅 − 𝐴)))
89	84, 88	breqtrd 4119	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < (𝑄 + (𝑅 − 𝐴)))
90	16	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑄 ∈ ℝ)
91	27	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝑅 ∈ ℝ)
92	91, 79	resubcld 8602	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (𝑅 − 𝐴) ∈ ℝ)
93	79, 90, 92	absdifltd 11801	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴) ↔ ((𝑄 − (𝑅 − 𝐴)) < 𝐴 ∧ 𝐴 < (𝑄 + (𝑅 − 𝐴)))))
94	78, 89, 93	mpbir2and 953	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (𝑅 − 𝐴))
95	23	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) / 2) ∈ ℝ)
96		avglt2 9426	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑄 < 𝑅 ↔ ((𝑄 + 𝑅) / 2) < 𝑅))
97	16, 27, 96	syl2anc 411	. . . . . . . . 9 ⊢ (𝜑 → (𝑄 < 𝑅 ↔ ((𝑄 + 𝑅) / 2) < 𝑅))
98	25, 97	mpbid 147	. . . . . . . 8 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) < 𝑅)
99	98	adantr 276	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → ((𝑄 + 𝑅) / 2) < 𝑅)
100	79, 95, 91, 81, 99	lttrd 8347	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 < 𝑅)
101	79, 91, 100	ltled 8340	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → 𝐴 ≤ 𝑅)
102	79, 91, 101	abssuble0d 11800	. . . 4 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑅)) = (𝑅 − 𝐴))
103	94, 102	breqtrrd 4121	. . 3 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) < (abs‘(𝐴 − 𝑅)))
104	67, 69, 103	ltapd 8860	. 2 ⊢ ((𝜑 ∧ 𝐴 < ((𝑄 + 𝑅) / 2)) → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))
105		apdifflemf.ap	. . 3 ⊢ (𝜑 → ((𝑄 + 𝑅) / 2) # 𝐴)
106		reaplt 8810	. . . 4 ⊢ ((((𝑄 + 𝑅) / 2) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (((𝑄 + 𝑅) / 2) # 𝐴 ↔ (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2))))
107	23, 1, 106	syl2anc 411	. . 3 ⊢ (𝜑 → (((𝑄 + 𝑅) / 2) # 𝐴 ↔ (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2))))
108	105, 107	mpbid 147	. 2 ⊢ (𝜑 → (((𝑄 + 𝑅) / 2) < 𝐴 ∨ 𝐴 < ((𝑄 + 𝑅) / 2)))
109	66, 104, 108	mpjaodan 806	1 ⊢ (𝜑 → (abs‘(𝐴 − 𝑄)) # (abs‘(𝐴 − 𝑅)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 ∈ wcel 2202 class class class wbr 4093 ‘cfv 5333 (class class class)co 6028 ℂcc 8073 ℝcr 8074 0cc0 8075 + caddc 8078 · cmul 8080 < clt 8256 − cmin 8392 # cap 8803 / cdiv 8894 2c2 9236 ℚcq 9897 ℝ⁺crp 9932 abscabs 11620

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-mulrcl 8174 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-0lt1 8181 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-precex 8185 ax-cnre 8186 ax-pre-ltirr 8187 ax-pre-ltwlin 8188 ax-pre-lttrn 8189 ax-pre-apti 8190 ax-pre-ltadd 8191 ax-pre-mulgt0 8192 ax-pre-mulext 8193 ax-arch 8194 ax-caucvg 8195

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-nel 2499 df-ral 2516 df-rex 2517 df-reu 2518 df-rmo 2519 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-if 3608 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-po 4399 df-iso 4400 df-iord 4469 df-on 4471 df-ilim 4472 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-frec 6600 df-pnf 8258 df-mnf 8259 df-xr 8260 df-ltxr 8261 df-le 8262 df-sub 8394 df-neg 8395 df-reap 8797 df-ap 8804 df-div 8895 df-inn 9186 df-2 9244 df-3 9245 df-4 9246 df-n0 9445 df-z 9524 df-uz 9800 df-q 9898 df-rp 9933 df-seqfrec 10756 df-exp 10847 df-cj 11465 df-re 11466 df-im 11467 df-rsqrt 11621 df-abs 11622

This theorem is referenced by: apdiff 16763

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